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GLfiEHUL
THE
WORKS OF ARISTOTLE
TRANSLATED INTO ENGLISH UNDER THE EDITORSHIP
OF
J. A. SMITH M.A.
FELLOW OF BALLIOL COLLEGE
W. D. ROSS M.A.
FELLOW OF ORIEL COLLEGE
PART 2 . DE LINEIS INSECABILIBUS
OXFORD AT THE CLARENDON PRESS
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THE
WORKS OF ARISTOTLE
TRANSLATED INTO ENGLISH UNDER THE EDITORSHIP
OF
J. A. SMITH M.A.
FELLOW OF BALLIOL COLLEGE
W. D. ROSS M.A.
FELLOW OF ORIEL COLLEGE
OXFORD AT THE CLARENDON PRESS
0fflfffl4i.
HENRY FROWDE, M.A.
PUBLISHER TO THE UNIVERSITY OF OXFORD
LONDON, EDINBURGH
NEW YORK AND TORONTO
PREFACE
It was the desire of the late Master of Balliol, Dr. Ben- jamin Jowett, as formulated in his will, that the proceeds from the sale of his works, the copyright in which he bequeathed to Balliol College, should be used to promote the study of Greek Literature, especially by the publication of new translations and editions of Greek authors. In a codicil to his will he expressed the hope that the translation of Aristotle's works begun by his own translation of the Politics should be proceeded with as speedily as possible. The College resolved that the funds thus accruing to them should, in memory of his services to the College and to Greek letters, be applied to the subvention of a series of translations of the works of Aristotle. Through the co-operation, financial and other, of the Delegates of the University Press it has now become possible to begin the realization of this design. By agreement between the College and the Delegates of the Press the present editors were appointed to superintend the carrying out of the scheme. The series, of which the first instalment is now brought before the public, is published at the joint expense and risk of the College and the Delegates of the Press.
The editors have secured the co-operation of various scholars in the task of translation. The translations make no claim to finality, but aim at being such as a scholar might construct in preparation for a critical edition and commentary. The translation will not presuppose any critical reconstitution of the text. Wherever new readings are proposed the fact will be indicated, but notes justificatory of conjectural emendations or defensive of novel interpretations will, where
193396
PREFACE
admitted, be reduced to the smallest compass. The editors, while retaining a general right of revision and annotation, will leave the responsibility for each translation to its author, whose name will in all cases be given.
Translators have been found for the Organon, Physics, De Caelo, De Anijiia, Historia Animalunn, De Animaliuni Generatione, Metaphysics, Eudemian Ethics^ Rhetoric, and Poetics, and it is hoped that the series may in course of time include translations of all the extant works of Aristotle. The editors would be glad to hear of scholars who are willing to undertake the translation of such treatises as have not already been provided for. and invite communications to this end.
J. A. S.
W. D. R.
March, 1908.
PARTS PUBLISHED
1. PARVA NATURALIA : by J. 1. Beare and G. R. T. Ross.
2. DE LINEIS INSECABILIBUS : by H. H. Joachim.
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DE LINEIS INSECABILIBUS
BY
HAROLD H. JOACHIM
FELLOW OF MERTON COLLEGE
^ OF THE
UNIVERSITY
OF
OXFORD
AT THE CLARENDON PRESS
1908
GENERAL
HENRY FROWDE, M.A.
PUBLISHER TO THE UNIVERSITY OF OXFORD
LONDON, EDINBURGH
NEW YORK AND TORONTO
OF THE
t'Tl L UNIVERSITY
INTRODUCTORY NOTE
The treatise Uepl arofjLaiv ypaixiJL(ov, as it is printed in Bekker's Text of Aristotle, is to a large extent unintelligible. But M. Hayduck, in the valuable paper which he contributed to the A^eue Jahrbiicher fiir Philologie und Paedagogik (vol. 109, part I, Teubner, 1874), prepared the way; and Otto Apelt, profiting by Hayduck s labours and by a fresh collation of the manuscripts, published a more satisfactory text in his volume Aristotelis quae ferunttir de Plantis^ &c. (Teubner, 1888). Many of the most difficult passages are discussed and elucidated in the prolegomena to this volume. Finally, Apelt included a German translation of the treatise in his Beitrdge ztir Geschichte der griechischen Philosophie (Teubner,
1891).
In the following paraphrase, I have endeavoured to make a full use of the work of Hayduck and Apelt, with a view to reproducing the subtle and somewhat intricate thought of the author, whoever he may have been. Though the treatise is published amongst the works of Aristotle, there are grounds for ascribing it to Theophrastus : whilst, for all we can tell, it may have been written neither by Aristotle nor by Theo- phrastus, but by Strato, or possibly by some one otherwise unknown. But the work — no matter who wrote it — is inter- esting for the close texture of its reasoning, and for the light which it throws on certain obscure places in Plato and Aristotle. Its value for the student of the History of Mathematics is no doubt considerable : but my own ignorance of this subject makes me hesitate to express an opinion.
I take this opportunity of thanking three of my friends, E. I. Carlyle (Fellow of Lincoln College) and A. L. Dixon (Fellow of Merton College) for their help in several of the mathematical passages, and W. D. Ross (Fellow of Oriel College) for his valuable suggestions, most of which I have adopted.
H. H. J.
January^ 1 908.
CONCERNING INDIVISIBLE LINES 968^
Are there indivisible lines? And, generally, is there a simple unit in every class of quanta ? ^
§ 1. Some people maintain this thesis on the following grounds : —
(i) If we recognize the validity of the predicates * big ' and ' great ', we must equally recognize the validity of their oppo- sites, * little ' and ' small '. Now that which admits practically an infinite number of divisions, is * big * not * little ' (or 'great* not 'small').^ Hence, the 'little' quantum and the 5 ' small ' quantum will clearly admit only a finite number of divisions.^ But if the divisions are finite in number, there must be a simple magnitude. Hence in all classes of quanta there will be found a simple unit, since in all of them the predicates ' little ' and * small * apply.
^ ^l. iv aTratTi rois noaois, and *8 ev aivacnv. The theory maintains that in dividing any quantum, of whatever kind, you will ultimately come to indivisible constituent quanta of the same kind. Every line, e.g., is com- posed of a finite number of indivisible lines : every solid of a finite number of indivisible solid constituents, i.e. solids not further divisible into solids. The advocates of this theory were feeling after the conception on which the differential calculus was based, and I presume that in the history of Mathematics they would take their place as the forerunners of Newton and Leibniz. Cf. Hegel, Wissenschaft der Logik^ vol. i. pp. 302-4.
*2. eWi ri dfxepes. I translate afxepes throughout by ' simple ', using 'simple' — in opposition to 'complex' or 'composite' — as equivalent to ' without parts '.
^ ^4 ff. TO TToXv and TO oXt'yoi'— that which contains many, and that which contains few, units— are the opposite predicates of discrete quanta, i.e. of Number (cf. Arist. Met. 992* 16, 17) : ro peya and t6 fiiKpov apply to continuous quanta. This at least seems to hold of the primary signification of these terms ; but the distinction is not maintained. Thus, e.g., in the CaUg. 4^ 20 ff., Number is instanced as a discrete quantum, Time and Surface are quoted inter alia as continuous quanta ; but tioKvi is predicated of Surface (5^ 2), and of Time (5*^ 3). I have added (or ' great ' not ' small ') in my translation, to complete the writer's thought. I do not suggest that there is an omission in the text.
^ »7. I translate e^^i Statpeo-eis throughout as 'admits divisions*, though at times the meaning of the Greek passes into 'contains divisions ': cf., e.g., 969*8.
968» DE LINEIS INSECABILIBUS
9 (ii) Again, if there is an Idea of line, and if the Idea is first of the things called by its name ^ : — then, since the parts are by nature prior to their whole, the Ideal Line must be indi- visible.^ And, on the same principle, the Ideal Square, the Ideal Triangle, and all the other Ideal Figures — and, general- izing, the Ideal Plane and the Ideal Solid — must be without parts : for otherwise it will result that there are elements prior to each of them. 14 (iii) Again, if Body consists of elements,^ and if there is nothing prior to the elements, Fire and, generally, each of the elements which are the constituents of Body must be indi- visible : for the parts are prior to their whole. Hence there must be a simple unit in the objects of sense as well as in the objects of thought.*
^ ^9, 10. 17 S' I8ea TrpaiTt} to)v (TWdivvyLUiv, i.e. the Idea is conceived as the limiting member of a series of things called by the same name and sharing the same nature in various degrees. Thus all lines, qud participating in the same linear nature, are called by the same name, ' line.' The Idea of Line is the Ideal Line which exhibits this linear nature perfectly and precisely : it is the limit from which actual lines derive, or to which they more or less approximate. If all lines were arranged in a series according to the degrees in which linearity obtained expression in them, the Idea of Line would be the first member of the series : it would be the Ideal Line which was just * Line ', neither more nor less.
' *ii. I accept Hayduck's conjecture abiaipiros, for the MSS. diaipfrrj, of which I can make nothing.
The theory contemplated by this argument is that in every kind of quantum — and, within spatial quanta, in every type of plane and of solid figure — there is an Ideal Quantum in the sense explained in the preceding note. This Ideal Quantum, it is argued, must be ' indivi- sible', i.e. simple. For, gud Ideal, it is the primary member in the series of which it is the Idea ; but, if it had parts, they would be prior to it, since the parts are prior to their whole.
^ *I4. €Tt el aciixaros ecm oToi;^€ta . . . Bekker. Read en el a-apaTos eari a-roixela, ' if there are elements of Body.' (The variant o-co/iara, though well attested, does not seem right.) a-Syfjia here, as the context shows, is not (as in 1. 13) mathematical solid, but perceptible or physical body.
* The first two arguments were directed to show that simple units are involved (i) in the Quanta of Mathematics, and (ii) in the Ideal Quanta postulated by a certain metaphysical theory. The present argument is intended to prove that the perceptible bodies (the bodies of Physics and of everyday life) ultimately consist of simple constituents. According to current views, all material things — ail ala-drjTo. aoofxaTa — consisted in the end of certain elementary constituents, viz. Earth, Air, Fire, and Water. An * Element ' means what is primordial, and therefore (it is argued) it must be without parts.
The writer does not explain to what precise form of physical theory he is alluding. He seems to be thinking of the somewhat vague and
SECTION I 968^
(iv) Again, Zeno's argument proves that there must be i8 simple magnitudes.^ For the body, which is moving along a line, must reach the half-way point before it reaches the end. And since there always is a half-way point in any ' stretch ' which is not simple, motion — unless there be simple magnitudes — involves that the moving body touches succes- 20 sively one-by-one an infinite number of points in a finite time: which is impossible.^
But even if the body, which is moving along the line, does touch the infinity of points in a finite time, an absurdity results. For since the quicker the movement of the moving body, the greater the ' stretch ' which it traverses in an equal time: and since the movement of thought is quickest of all 25 movements : — it follows that thought too will come succes- sively into contact with an infinity of objects in a finite time. 968*^ And since ' thought's coming into contact with objects one- by-one' is counting, we must admit that it is possible to count the units of an infinite sum in a finite time. But since this is impossible, there must be such a thing as an 'indivisible line*.^
popular view, which regarded Earth, Air, Fire, and Water as the ' Letters * of the Alphabet of Reality, and the physical universe as a complex of * Syllables ' and ' Words * in which these four Letters are variously com- bined. But the principle of the argument would apply to the more refined forms which the theory assumes in the Timaeus of Plato and in Aristotle's physical writings. The primordial triangles of the Timaeus^ qua Elements of all bodies, are presumably without physical parts, i.e. physically indivisible. And the Earth, Air, Fire, and Water, which (according to Aristotle) are the chemical constituents of all ofioiofxeprj — and therefore the primary constituents of all composite bodies — , are ' to. drrXa a-wfiaTa*, although the character of each of them is dual, i.e. is exhibited in two of the four fundamental qualities. (For Aristotle's theory of the Elements, cf. my article on 'Aristotle's Conception of Chemical Combination', Journal of Philology y No. 57.)
^ ^19. avayKx) ti fxeyedos afifpis elvaij i.e. there must be such a thing as a simple magnitude. For Zeno's argument cf. Arist. Phys. 187* i and Simplicius ad loc.
^ * 1 8-23. Here and elsewhere I have not scrupled to paraphrase rather freely, in order to bring out the argument. From the infinite divisibility of the continuous path of the moving body, Zeno concluded that motion was impossible ; for the moving body would have to come successively into contact with an infinite number of points in a finite time. The advocates of 'simple units' argue that, since ?notion is a fact, the con- tinuous path cannot be 6.W\s\h\t ad injinitum : i.e. any given line must consist of a finite number of ' indivisible lines '.
^ ^4. The Greek is drf au us arofios ypamxr]. The meaning here (as in
B 2
968^ DE LINEIS INSECABILIBUS
^ (v) Again, the being of * indivisible lines ' (it is maintained) follows from the Mathematicians' own statements. For if we accept their definition of ' commensurate ' lines as those which are measured by the same unit of measurement,^ and if we suppose that all commensurate lines actually are being measured,^ there will be some actual length, by which all of them will be measured.^ And this length must be indivisible. For if it is divisible, its parts — since they are commensurate with the whole — will involve some unit of measurement lo measuring both them and their whole. And thus the original
968'^ 5 : cf. also 968^ 19) cannot be given by the English 'there must be an indivisible line ' or ' a line which is indivisible '. We must translate either as above, or by the plural ' there must be indivisible lines '.
The argument (^23-^4) is directed against a particular view of thought and of counting. * Assume ' — the writer says in effect — ' that the moving body does in fact touch an infinity of points one-by-one in a finite time. According to your view that thought is the quickest of all movements, it will follow a fortiori that thought touches an infinity of objects one-by-one in a finite time: i.e. (according to your definition of counting) that we can count an infinite number in a finite time. But this is impossible. And the only way to avoid this absurdity, whilst recognizing the fact of motion^ is to postulate " indivisible lines ".'
The theory that thinking is a movement of the Soul was not held by Aristotle : for he argues in the de Anima (A. ch. 3) against all attempts to define the Soul as 'that which moves itself,' and maintains that *it is impossible that movement should be a property of the Soul' (1. c. 406* 2 ff.). Certain speculations of Plato in the Timaeus (which Aristotle criticizes, 1. c. 406^ 26 ff.) regard thought as a movement: and Theophrastus and his pupil, Strato, are known to have maintained that — ^ thought was a movement of the Soul (cf. Apelt, Beitrage &c., p. 270). But we must not infer — as Apelt (I.e.) does — that Aristotle is not the author of the present treatise : still less that it was written by Theophrastus or Strato. For we are here dealing with an argtmientum ad homine?n, and the writer is not himself committed to the view that thought is a movement of the Soul.
* Cf. Euclid, Elements^ Bk. X, def. i ^vfijxeTpa fxeyedrj Xcyerai to. rco aiTto iJi€Tp(o fieTpovfxcva, ....
* ^6, 7, reading (with all the MSS., except N) el aufxfxfTpoi elaip al t<o avT<i fi€Tp<o perpovpevai, ocrai 8' fieri (Tvpp,€Tpoi, Traarai etcri p-erpovpevai.
Apelt in his text followed N, and read oo-at S' iWi fxerpovpemi, naaai flat <Tvp.p.fTpoi. But in his translation he reverts to the best attested reading.
I substitute a comma for Bekker's colon after fxerpovfievai in 1. 6, because the whole clause is dependent on el. The logic of the passage is, * If we accept X, and combine with that the supposition j, there must be indivi- sible lines: for on those suppositiojts there will be a unit length which must be indivisible.'
" ^8. CO Trao-at perprjOrjaovTai, ' whereby all commensurate lines will be measured': but, as appears from 969^ 10-12, the argument (by a some- what transparent fallacy) regarded all lines as 'commensurate'. See next note.
SECTION I 968^
unit of measurement would turn out to be twice one of its parts, viz. twice its half.^ But since this is impossible, there must be an indivisible unit of measurement.^ And just as all the lines, which are compounded of the unit, are composed of * simples ', so also the lines, which the unit measures once, consist of ' simples '.^
And the same can be shown to follow in the plane figures too. For all the squares, which are drawn on the rational 15 lines, are commensurate with one another ; and therefore (by the preceding argument) their unit of measurement will be simple.'^
^ ^10, II. Bekker reads more fiepovs rivos elr] [uvul W*] BLTrXacria [8nr\a(Tiav rc N, 8inXd(Tiov LW*] rfip fjni(T€iav, .•. . From the reading of LW*, I suspect that the author wrote dirrXaaiayv (cf. e.g. Euclid, Elements^ -^ Bk. X, prop. 9 : the word occurs in [Arist.] Probl. 923* 3, De Mimdo, 399a 9). In place of rqv rjiucreiap, Z* apparently (' ut videtur', Apelt says in his apparatus criticus) reads ri}? fjfxia-vos. Hayduck conjectured &aT€ ^erpov av eir] dnr\n(ria rrjs fjnia-eias, or axTTf ixeTpelv tiv cltj 8in\acria rfjv rjfxia-etav. Apelt suggests coo-re fiepovs Tivos av elvai dtTrXnaiav tt)V rjfiiaunv, but I do not see that this is of much assistance. I have translated as if the text were (oare fiepovs nvos (^^v} ftrj dtirKaa-LQiv, ttjs rjixiaens' enfl de ktX. But it is possible that ttjs r)fii(rens ought to be excised as a gloss explanatory of pepovs Tivos.
It appears (from the criticism of this argument at 969^ 10-12) that the advocates of 'indivisible lines' reasoned thus: — 'Lines measured by the same unit are "commensurate". Now take any line, ^i5. It will always be possible to find, or draw, a line containing without remainder a multiple of the units in AB: i.e. AB will be "commensurate". Let then all '^ commensurate " lines (i. e. a// lines) be actually measured. There will be an actual length, or infinitesimal line, xy., which is the unit of measurement of them all. And xy must be indivisible. For, if not, xy will have parts : and thus the unit will be multiple (v. g. will be twice its own half), which is absurd.' The fallacy is obvious, and is exposed at 969** 10-12. Any line AB can become 'commensurate' with some line: but, because commensurate with so77ie line, it is not necessarily commen- surate with all lines, or ' commensurate ' absolutely. One would indeed think the fallacy too obvious to have been committed : but, in the refutation, the writer refers to it as a ridiculous and obvious sophism, cf. 969^ 6-10 and 12-15.
^ ^11. The MSS. read ^/ztWtai/, eVetSi) rovr ahvvarov av cirj pirpov.
I read with Apelt incX di tovt ahvvarov, (aStaipfTOv) av etrj p.irpov, and place a colon before eVe/. The insertion of ddiaipfrov was suggested by Hayduck, after the Latin translator, Julius Martianus Rota, who writes ' quoniam vero hoc fieri nequit, indivisibilis esse mensura debet '.
^ ^12-14. Let xy be the unit of measurement, which measures all com- mensurate (i.e. all) lines. Then all lines will ' consist ' of simples : for they will either contain xy once, or more than once, without remainder.
* ^14-16. The object of this argument is to show that 'simple units' must be admitted in plane figures, as well as in lines. The writer selects the square as an example of plane figure, and maintains that all squares
96S^ DE LINEIS INSECABILIBUS
1 6 But if {per impossibile) any such unit- square be cut along any prescribed and determinate line, that line will be neither ' rational ' nor ' irrational ', nor any of the recognized kinds of (irrational) lines which produce rational squares, such as the 'apotome* or the 'line ex duobus nominibus '. Such lines,
20 at which the unit-square might be divided, will have no nature of their own at all ; though, relatively to one another, they will be rational or irrational.^
consist ultimately of a finite number of minimal squares, not themselves divisible into any smaller plane figures.
In order to understand the argument, and the fallacy on which it rests, it will be necessary to explain certain technical terms of Greek geometry. (l) The expression ra dno t5>v prjrwv ypafifxap (1. 15) must — in accordance with Euclid's invariable usage — mean ' the squares on the prjTnl ypanfiai \ The noun implied is Tfrpdyava: but to dno followed by the genitive is constantly used without rerpdyavov, and always means the square on such- and-such a line. (Hence Apelt is wrong in translating *Alle Fldchen mit rationalen Seitenlinien '.) (2) The proper meaning of pr]rai ypap.fiai will be seen from the following definitions of Euclid (E/em. X) : — d^f. 3 ' . . . given any straight line, there are an infinity of straight lines commen- surate with it and an infinity incommensurate with it— incommensurate either in length only, or both in length and in respect to the areas which they and it produce if squared (at \xkv fxljKd p,6vov, al de koi dvpafiu : see below). Let the given straight line, and all the straight lines which are commen- surate with it (whether commensurate both fjiT)Kei and 8vvdnei,or dwdfxei only), be called " Rational " {pT]Tm) : and let the straight lines, which are incom- mensurate with it, be called *' Irrational " (dXoyoi) ' : def. 4 ' And let the square on the given straight line, and all the squares commensurate therewith, be called " Rational " : and let the squares incommensurate with it be called "Irrational" . . .' (3) Any straight lines, which are multiples of the same unit of length, are said to be (rvfifierpoi p^Kei. If e.g. the unit of measurement be rj nodLaia (the line one foot long), all lines con- taining a whole number of feet are a-vpixfTpoi prjKu. But lines which do not contain a whole number of the same unit of length are said to be avfififrpoi Svvdpei, if they form squares containing a whole number of the same unit of area. All lines, which are arvpfifTpoi fxrjKfi, are necessarily also (Tvpp.eTpoi Swdfifi — but the converse does not hold (Eucl. E/em. X, prop. 9, Coroll.).
We are now in a position to understand the argument of ^14-16. The writer extends the relative term ' rational ' illegitimately (making it absolute), just as before he illegitimately extended the relative term ' commensurate '. All ' rational ' lines are by definition Swdpei a-vpfieTpoi : and therefore all squares on rational lines are commensurate. And if we suppose them actually measured, there will be an actual minimal square, the unit of measurement of them all (cf. above, 968^ 6-8) : and this minimal square can be shown to be indivisible — i.e. not to contain smaller plane figures — as before the unit-line was shown to be ddiaiptTop (968^ 8-12). But— tmless we assume that all lines consist of i7idivisible and equal unit' lines — we cannot assume that all lines are ' rational ' in Euclid's sense, nor that all squares are commensurate with one another.
* ^16-21. The text of this passage is corrupt, and the argument obscure, and I have no confidence in the interpretation which I have given. As
SECTION II 968''
§ 2. To these arguments we must make the following 21 answers : —
(i) (a) In the first place, it does not follow that the quantum, which admits an infinite number of divisions, is not ' small ' or * little '. For we apply the predicate ' small ' to place and magnitude, and generally to the continuous (and in some quanta the predicate ' little ' is suitably applied) ^ ; and nevertheless
regards the text, I adopt Apelt's reading in 1. 19, S>v dvm^eis prjrai^ olov anoTOfif) T) fj eK dvolu ovofidroiv for the MSS. S}V 8f) vvv \vvv 8tj N] etprirai, olov a.7roTOfir)v €k dvolv ovofiaTOiv.
The lines called (k dvo'iv ovoiidroiv and aTroTofiT] are two types of Irrationals (i.e. fir]K€i da-vufxfrpoi, but dvvdfxei (rvpfifTpoi) which play a large part in Euclid, Elem. Bk. X.
The line e/c hvo\v ovofxdroiv is defined in Prop. 36 thus : — ' If two rational straight lines, which are commensurate bwdixei only, be added together, the whole line is irrational : let it be called
" the line €< 8vo ovofidrav " ' : — i.e. the >\ q q
line AC \s that type of ' Irrational '
(irrational relatively to AB and BC) which is called ' ex duobus nomi- nibus', if it is such, that AB^ is commensurate with BC^, but AB is incommensurate with BC. AB and BC are called the 'ovouara' of AC.^
The dTTOTopr] is defined in Prop. 73 thus :— * If from a rational line there be taken a rational line commensurate with the whole line dvudfxei only, the remainder is irrational: let it be called an " aTroro/ijJ " ' : - i.e. if the
line AB be divided at C, so that AB^
is commensurate with CB"^ but AB is ^ q q
incommensurate {mkh) with CB^ then
AC '\s called an dnoTopir]. The complementary part of the whole line (viz. CB) is called relatively to AC its TrpoarapfidCova-a (cf. Propp. 79-84). We might illustrate these two types of ' Irrationals' thus: — (i) Let the two oi/d/iaTa be I and ^/^. Then the whole line, AB + BC, = (i + a/$). I is incommensurate with \^s, but_(i)'^ and i\/s)^ are commensurate. (2) Let the whole line, AB, be ^5. Divide AB at C, so that CB=i. Then {\/sf is commensurate with (i)^, but \/s is incommensurate with i. ^ C (the aTTOTon^) = ( yj - 1 ) .
I have interpreted the argument (968^ 16-21) as a reductio ad absiirdum. ' Suppose,' the writer urges, * the unit-square is divided. The line dividing it will not answer to any known line : i.e. there is no line recognized by Geometry at which the unit-square could be divided into smaller plane figures. For whatever line of division be selected, that line will neither be rational nor irrational : nor will it fall under any of the recognized types of line which, though irrational qua lines, produce rational squares, or otherwise exhibit relations studied by Geometry. Any such lines of division will, in fact, belong to a new order of lines, which may be expressed as rational or irrational in terms of one another, but not in terms of the ordinary geometry of lines.'
^ ^24. Kai e(^' lav \ikv dpfioTTei to oXiyov . . .
Hayduck suggested koi e(^' oov dpfioTTd, oXiyop, which would be an improvement, though the excision of fiev seems unnecessary. (It is,
968*' DE LINEIS INSECABILIBUS
we affirm that these quanta admit an infinite number of divisions. 25 (i) (b) Moreover, if in the composite magnitude there are contained {indivisible) lines/ the predicate ' small ' is applied to these indivisible lines, and each of them contains an infinite 969^ number of points. But each of them, qud line, admits of divi- sion at a point, and equally at any and every point : hence each of these indivisible lines would admit an infinite number of divisions just like the non-indivisible lines.^ Moreover, some amongst the non-indivisible lines are 'small'. But every non-indivisible line admits of division in accordance with any prescribed ratio : and the ratios, (in accordance with which any such line may be divided), are infinite in number.^
however, omitted by Z*.) Apelt defends the MSS. reading, but interprets Kot €<^' 2)v — oXt'yoi/ as part of the subject of the sentence, taking \iiKp6v as predicate of the whole. This seems difficult, because {a) the [ikv [((f) o)v fxh] is purely gratuitous, and (d) there is no reason why the writer should over-ride the distinction between fxiKpov and 6\iyov.
If the TO be retained, the clause must, I think, be treated as parenthetical and interpreted as above.
^ ^25. Apelt reads (with N) eVi 8' el iv rois o-vfifieTpois ypafijjiai elai ypaixixai. He suggests that the passage ought to be emended to run €Tt fi' ci fuiais (TvixfiiTpoi ypnfjLfjiais elal ypafifiai, Kara tovtcdv arofiov Xtycrat to fxiKpov, Of this I can make nothing : nor do I see how he could defend his translation ' und von ihrem Mass gesagt wird, es sei unteilbar klein '.
All the MSS. (except N and Z*) read ert 5' el iv rep avv6eT(o ypanfiai, Kara TovTcov tS)v [tS)v omit t. NPW^Z*] ard/io)!/ kt\. But as in PW* and L there is a lacuna after a-vvdera, I have ventured to conjecture ert 5' el iv tw avvdeT<o (^uTOfioi €to-i) ypafxfiai . . . With avvOtTca I understand fieyidei or
^ ^2, 3. The text in Bekker is koI ofxoias KaO' OTroiavovv aneipovs av e'xoi diaipeaeis anatra rj p.rj uTofios. I follow Apelt in placing a colon after oTToiavovv, and in reading aneipovs ovv e'xot . . . After an-aa-a I insert av, &)f, combining the readings of NZ* and H*. The passage then runs KaB^ OTTOiavovv' aneipovs ovv e^oi biOLpecreis anacra av, a>s rj p.r) uto/xos.
' ^3-5. The text given in Bekker is eviai 8e tovtoav els fioKpa [piKph LZ*, p.iKpav NH*] Ka\ aneipoi 01 Xdyoi [for ol Xd-yoi LFH^W* read oXiyov, Z* reads koI oXiyov]. nacrav 8e Tp.r]6rjvat tov iniraxOevTa SvvaTov ttjv fif) aTofiov. [For Trjv fxr] H.^ reads TOfxfjv ttjv, and N places tov innaxdevTa after aTOfiov.]
I have ventured to read evtat 8e tovtmv elal fxiKpai' naa-av Se TfiTjOijvai dvvaTov Trjv p-rj arop-ov tov [? Kara rov] iniTaxBevTa Xoyov' Ka\ aneipoi oi Xoyoi. If this be thought too bold, we might retain the MSS. order, and read evtai . . . piKpai' koi aneipoi oi Xdyoi, nacrav . . . ttjv p.f} aropov tov invraxOevTa. We must then take koI aneipoi oi Xoyoi closely with the following words. The only authority for Xoyov (which Apelt inserts after iniTaxOevTa) is the editio princeps.
SECTION II 969'
(i) (c) Again, since the * great ' is compounded of certain 5 * smalls ', the * great ' will either be nothing, or it will be identical with that which admits a finite number of divisions.^ For the whole admits the divisions admitted by its parts: i.e. its divisions are finite or infinite, according as their divisions are finite or infinite.- It is unreasonable that, whilst the small admits a finite number of divisions only, the great should admit an infinite number ; and yet this is what the advocates of the theory postulate.'^
It is clear, therefore, that it is not qud admitting a finite and 10 an infinite number of divisions that quanta are called ' small ' and ' great ' respectively. And to argue that, because in numbers the ' little ' number admits a finite number of divi- sions, therefore in lines the ' small ' line must admit only a finite number of divisions, is childish. For in numbers the more complex are developed out of * simples \ and there is a determinate something from which the whole series of the numbers starts, and every number which is not infinite admits 15
The argument of the whole passage (968^ 25-969^ 5) I take to be as follows : — ' Every composite length contains lines. According to the theory, some amongst these lines are "indivisible". But every one of these lines, qtid line, contains an infinity of points, and admits therefore an infinity of divisions ; for a point is that at which a line can be divided. Yet by comparison with the whole (com- posite) length, all the "indivisible" lines, and at least some of the divisible lines, are "small". Hence infinitely-divisible quanta may be "small".'
The Xoyoi (969* 4) are, I presume, the numerical ratios in which any line may be divided. ^
^ *7. 1 accept Apelt's conjecture ro fieya for the MSS. ov fieya.
^ *8. TO yap oXop ras tmv {xeprnv (^€1 diaip€(r€ts 6p,oici)S, i. e. the divisions
which the whole admits— since it is the sum of its parts— are the sum of the divisions which the parts admit, and the number of divisions is either finite or infinite in both cases. The argument, to which this is a reply, assumed that the large number of divisions in the ' great ' was ' practically infinite ' (968* 4), whilst the * small ' admitted only a finite number of divisions.
^ Reading, with Apelt, aXoyov (for the MSS. (vXoyov) in ^8, and ovtm S' a^wva-tv (for the MSS. ovrcos- d^ioxja-iv) in *IO.
It is just possible, however, to retain the MSS. reading, if we con- strue d^iovaiv as dative plural of the participle, and remove the stop before ovtojs. 'And yet it is a reasonable inference for them, with their assumptions, that the "small" admits a finite number, and the "great" an infinite number, of divisions': — i.e. the view in question has just been shown to be false, but it follows plausibly enough from their premisses.
969^ DE LINEIS INSECABILIBUS
a finite number of divisions ; but in magnitudes the case is not parallel.^
17 (ii) As to those who try to establish the being of the indivi- sible lines by arguments drawn from the Ideal Lines, we may perhaps say that, in positing Ideas of these quanta, they are assuming a premiss too narrow to carry their conclusion ; and, by arguing thus, they in a sense destroy the premisses which they use to prove their conclusion. For their arguments destroy the Ideas.^
21 (iii) Again, as to the corporeal elements,^ it is childish to postulate them as ' simple '. For even though some physicists do as a matter of fact make this statement about them, yet to assume this for the present inquiry * is a petitio principii. Or rather, the more obviously the argument would appear to involve a petitio principii^ the more the opinion is confirmed that Solids and Lengths^ are divisible in bulk and distance.^
^ The above arguments, from 968^ 21, are directed against the first argument (968* 2-9) of the advocates of indivisible lines.
'^ ^17-21. This is directed against the second argument (968*^ 9-14) of the advocates of indivisible lines.
Karao-Kcudjo) is used in the sense of ' establishing ' (e. g. a conclusion or a definition) in opposition to di/ao-Keuafo), 'to overturn ' : cf. e.g. Pr. Anal. 43* I, Top. 102* 15, &c. The argument in question aimed at proving the universal affirmative that all lines contain indivisible lines as ultimate constituents. And it tried to base this conclusion on the indivisibility of the Idea of line, i. e. it involved the assumption of Ideas of quanta, or at least of Ideas of lines. But from what holds good of Ideal lines, you can make no valid inference to all lines : the premiss is particular (Ideal Lines, i. e. some lines, are indivisible), and cannot serve as the basis of the universal conclusion which is to be proved.
Moreover, it is dangerous for the advocates of Ideas to use an argument of this kind. For their opponents may retort that, if the assumption of Ideal quanta leads to the absurdity of indivisible lines, then so much the worse for the Ideal theory. In the sphere of mathematics, they may say, the assumption leads to consequences mathematically absurd ; hence the whole theory of Ideas is discredited.
^ *2i. TrdXti/ 5e rcoj/ aioixariKav (TToixelciv . . . The genitive alone seems impossible. I read ndXiv 6' eVt rav kt\. (coll. 969^ 6).
* ^23. npos ye rrjv vTroKcifxevrjv (XK^-^iv ... I can find no exact parallel to this use of vnoKeifievrjv, but cf. perhaps Pol. 1331^ 36. In the next two lines o(r(o fxaWov . . . too-w fiaXXov is an expression without parallel in Aristotle.
^ ^26. Reading aSifxa koX firjKos, and interpreting arS>ixa as ' geometrical solid ' (not as ' perceptible body '). The difficulty in this reading is that Koi Tols oyKois Kai vols diacTTTjfxaaiv ought to mean ' both in bulk and dis- tance': but this would be true of aafxa only. Disjunctively, of course, it is true of (ra>fxa and firjKos, but the double Kai is certainly awkward. Apelt in his translation adopts the reading of LNH^W* o-oj/xa fxrjKovs : but he
SECTION II 969*
(iv) The argument of Zeno does not establish that the 26 moving body comes into contact with the infinite number of points in a finite time, if the period and the path of the motion are considered on the same principle.^ For the time and the length are called (both) infinite and finite (from different points of view), and admit of the same divisions (if considered both on the same principle).^ 30
Nor is * thought's coming into contact with the members of an infinite series one-by-one ' countings even if it were supposed that thought does ' come into contact ' in this way with the members of an infinite series. Such a supposition perhaps assumes what is impossible: for the movement of thought does not, like the movement of moving bodies, essentially involve continua and substrata. 969^
If, however, the possibility of thought moving in this fashion be admitted, still this moving is not * counting ' ; for counting is movement combined with pausing.
It is absurd — we may perhaps suggest to our opponents —
can only translate this by making the fiaWov of 1. 25 do double duty. All would be plain if we could omit /cat iirjKos altogether, and read o-w/xr [i. e. ' perceptible body '] koI t. oyKois k. t. diaa-Trjfiaa-iv.
* ^21-26. This paragraph is directed against the third argument (968* 14-18) of the advocates of indivisible lines. That argument rested on the assumption that perceptible bodies involved Elements, i. e./r/>//rtry constituents. Even admitting that some physicists speak in this way about the constituents of bodies, to take this as a premiss to prove that there are indivisible magnitudes is to beg the question. (Cf. Hayduck, 1. c, p. 163, for the above interpretation.) Or at least it looks like beg- ging the question; and the more it looks so, the more the prevailing opposite opinion is confirmed. For a view gathers strength in proportion to the weakness of the arguments advanced against it.
^ ^27, 28. The MSS. read .... avix^i^d^ei ov arvfnrenfpaa-^eva) XP"^¥ '"'^'^ aTTftpcoy anTcrai [LNPW^Z* : aima-Bai cetert\ to <pep6n€Pov q)8i tov avrov
TpOTTOV.
Bonitz conjectured to iv TrcnepiKTfiePM xpova . . . anTca-Oiu. I read with Apelt (Bs iv nenepaafxivco . . . anTCTai. And in I. 30 I accept Apelt's to? avTus e'xei diaipeaeis (for which he compares Arist. Phys. 235* 15) for the MSS. Touai^ or TO(Tavra^^ 6;^fi diaipiafis.
2 a26-3o. The period and the path of the motion, gud continuous quanta, are divisible ad mjinitimi : but, qua determinate (finite), may both be regarded as containing a finite number of units, i.e. as admitting a finite number of divisions only. Zeno's argument depends on the fallacy of viewing the period as finite, and neglecting its divisibility ad infinitum qua continuous ; whilst the path is viewed {qua continuous) as an actual infinity of points, and its finiteness is neglected. [Cf. also Aristotle's solution of Zeno's argument, Phys, 233* 8-34.]
969^ DE LINEIS INSECABILIBUS
that, because you are unable to solve Zeno's argument, you should make yourselves slaves of your inability, and should commit yourselves to still greater errors, in the endeavour to support your incompetence.^
6 (v) As to what they say about ' commensurate lines ' — that all lines, because commensurate ^, are measured by one and the same actual unit of measurement — this is sheer sophistry ; nor is it in the least in accordance with the mathematical assumption as to commensurability. For the mathematicians do not make the assumption in this form, nor is it of any use to them.
lo Moreover, it is actually ^ inconsistent to postulate both that every line becomes commensurate, and that there is a common measure of all commensurate lines.^
^ This and the preceding argument are directed against the fourth argument (968* 18-^4) of the advocates of indivisible lines.
The writer urges (i) that Zeno's argument involves a fallacy, which the advocates of indivisible lines have failed to detect (969* 26-30). (ii) That the movement of thought (' psychical process ') is not analogous to the movement of a body. The latter is essentially conditioned by the con- tinuity of the path traversed and the continuity of the body moving : for physical movement takes place in a material substratum— \.^. a solid material body— and along a path in space, (iii) That if the movement of thought were analogous to the movement of a body, more than this would be required to constitute ' counting '. For to ' count ' is not merely to - — traverse a continuous path, coming into instantaneous contact with the infinite succession of points, into which that path may be mathematically resolved : to ' count ' essentially involves pausing at the successive steps of the process, (iv) That the argument drawn from ' counting ' is an extravagant supposition by which the advocates of ' indivisible lines ' are endeavouring to support themselves in an erroneous position— a position really due to their incompetence in failing to detect Zeno's fallacy.
^ "7. The M SS. read o)? on al rraaai. This presumably means ' e. g. that ' or ' viz. that '. But it is very doubtful whether o)? on could be used in this way as equivalent to the ordinary olnp on. I propose to read w?, on (^(TVfxfKTpoiy, al Traaai ....
■' Kal ivavriov.
* ^6-12. This is directed against the fifth argument of the advocates of indivisible lines (cf. above, 968^ 4-14)-
It is difficult to be sure of the meaning of 969^ 10-12, owing to the obscurity of the argument which is being attacked. I think the point of the criticism is as follows. The mathematical definition of commensurate lines can always be satisfied, in the sense that, given any line AB^ you can always find a line ' commensurate' with it : i.e. any line can become ' commensurate ' with some line. But though all lines are * commen- surate ' in this sense, they are not all commensurate with one another, and have not got one and the same common measure. Yet the advocates of ' indivisible ' lines maintain both (i) that any line can become ' commen-
SECTION II 969*
Hence their procedure is ridiculous, since, whilst professing 12 that they are going to demonstrate their thesis in accordance with the opinions of the mathematicians, and by premisses drawn from the mathematicians' own statements, they lapse into an argument which is a mere piece of contentious and sophistical dialectic — and such a feeble piece of sophistry too ! For it is feeble in many respects, and totally (unable) to escape paradox on the one side, and destructive scientific criticism on the other.^
Moreover, it would be absurd for people to be led astray by 16 Zeno's argument, and to be persuaded — because they cannot refute it — to invent indivisible lines : and yet to pay no atten- tion to all those theorems concerning lines, in which it is proved that it is impossible for a movement to be generated such that in it the moving thing does not fall successively on each of the intervening points before reaching the end-point. For the 25 theorems in question are far better established, and more generally admitted, than the arguments of Zeno.^
surate ', a?id (ii) that all commensurate lines have a common measure : and these two propositions are inconsistent. For (i) is true only if ' commensurate ' be used in a 7'elative sense ; and then (ii) is false. Whilst (ii) is true only if ' commensurate ' be used in an absolute sense ; and then (i) is false.
^ ^12-16. Bekker reads worf -yeXoioi/ to \rQ om. W*] Kara [/cal N] ra? (Keipciv do^as Kal i^ hv avToi Xtyovai cfydaKovres Sei^eir, its ipUTTiKov afxa Koi ao(^i(TTi,Kbv cKKXipeiv [eyKXiPfiv LPW* ey/cXiVat N] \6yov, kq\ ravd^ ovtcos daOep^. TToWaxfi [iroWaxats LPW*] yap dadeprjs eort Koi rravra rponov 8ia(jivyeiv koi to. napado^a kol tovs eXcyxovs.
By reading (pdaKopms in 1. 13 very tolerable sense may be made of the first sentence. Apelt follows N and reads to kuI tus ktX . . . eyKXivai . . . * ridiculum est et illorum (sc. mathematicorum) placita et ea, quibus ipsi argumenta sua superstruunt, in sophisticas captiones detorquere.' But avToi (cf. 968^ 4, to which this refers) is most naturally taken as ' the mathematicians': and in any case Apelt's interpretation is not con- vincing.
The last sentence seems to be corrupt. The general sense of the passage would be satisfied by ndpTa Tponop ddvuaTOS (or ddvparei) dia(f>vye'ip . . . : but I hesitate to propose any reading. The point seems to be that the advocates of indivisible lines are exposed to a double fire. They are using as an argument what to common sense is ridiculous paradox, and what to professional mathematicians is demonstrably unscientific.
^ ^16-26. In the above paraphrase I think I have reproduced the general drift of this passage. Zeno showed that if a body is to move from A to B, it must touch all the intermediate points before reaching B: i.e. it must traverse an infinity in a finite time. And he argued that motion is impossible. The advocates of indivisible lines replied : ' Motion is a
969^ DE LINEIS INSECABILIBUS
26 § 3. It is clear, then, that the being of indivisible lines is neither demonstrated nor rendered plausible — at any rate by the arguments which we have quoted. And this conclusion will grow clearer in the light of the following considerations : —
29 (A) In the first place,^ our result will be confirmed by reflec- tion on the conclusions proved in mathematics, and on the assumptions^ there laid down — conclusions and assumptions
fact, and therefore— since Zeno's argument is sound— the line AB must consist of a finite number of indivisible unit-lines.' The writer here rejoins : * Geometry proves that there can be no motion without the phenomenon to which Zeno called attention. A motion, such as your theory requires— a motion in which the moving body does not traverse successively all the intermediate points— does not, and cannot, occur. And the theorems, in which geometry establishes this, are far more con- vincing than the arguments of Zeno.'
In other words : — Geometry, assuming motion to be a fact, shows that the moving thing does traverse an infinity of intervening points, and shows that there can be no motion in which this does not take place. The advocates of indivisible lines have made no attempt to refute these geometrical proofs. Their postulate of 'indivisible lines', even if it evaded Zeno, collides with these far more solid facts of geometry : for the kind of motion which v/ould occur, if there were indivisible lines, is shown by geometry to be impossible.
The text of this passage is so corrupt that it seems hopeless to make out the details of the argument.
In 11. 19-21 the writer is clearly referring to the movement of a straight line about one of its terminal points, whereby a semicircle (and, ultimately, a circle) is generated. duiaTtjixa is the regular term in Euclid for the distance at which, from a given point as centre, the circumference of a circle is drawn. Cf. e.g. Eucl. Elem. I. 22 . . . Kevrpa ^lev rw Z, diaarrjixaTi Se ra ZA kvkXos yfypd(j)6a) 6 AKA . . ., and SO constantly. {hidaTTjixa in fact = * radius'.)
In 1. 19 we should read with Apelt 8m de^rffv} rrjs evOeias fls to fjfiiKvKKiov [so NZ* : the other MSS. read fjfiwXiov] Kivr)(nv^ ...
But Apelt (in the Prolegg. to his text) proposes other emendations for the rest of the passage, which are not convincing. It is best to recognize that the passage is hopeless, until somebody can discover the exact geometrical theorems to which the writer is referring.
^ ^28 ff. The writer is going to show that the doctrine of indivisible lines cannot be reconciled with mathematics. It collides with the con- clusions established in mathematics, and it collides with the premisses laid down by the mathematicians. He adduces a series of instances of such collision, and sums up at 970* 17 ahXa 8' av rtr Kai ertpa Toiavra avmyoi' naai yap cds ^Inelv epavTiovrai toU (V toIs na9^p.a(riv.
7rpa>Tov p.€v (^29) is answered by ttoKiv (970* 19).
^ ^30. I have translated ndepivcop ' assumptions '. It probably includes (a) definitions of the meaning of ' Subjects ' and * Attributes ' (= Aristotle's opia-fios, where that is used in a restricted sense and contrasted with vrroBeais: cf. e.g Pos^. Anal. 72* 21-24), and {b) Aristotle's tStai apx"*> i.e. definitions of the meaning of the 'Subjects' accompanied by the viToOfaii oTi (an (cf. e.g. Pos/. Anal. 76* 32-36).
SECTION III 969*
which we have no right to reject except on more convincing arguments than those adduced by the advocates of indivisible lines.^
For (i) neither the definition of ' line ', nor that of ' straight line ', will apply to the indivisible line, since the latter is not between any terminal points, and does not possess a middle.^
(ii) Secondly, all lines will be commensurate. For all lines 97©° — both those which are commensurate in length, and those which produce commensurate squares — will be measured by the indivisible lines.
And the indivisible lines are all of them commensurate in length (for they are all equal to one another), and therefore also they all produce commensurate squares. But if so, then the square on any line will always be rational.^
* ^30, 31. I read with Apelt (after NZ*) a dUaiov^ fxevew fj ma-TOTepois \6yois Kiveiv.
Since obviously the mathematician adduces no arguments in support of his Tidefievn^ I have interpreted maroTepois as above. (It is possible, however, that we should translate ' more convincing than the mathe- matical statements ' : cf. de Caelo 299* 5 Kalroi dUaiov r/v fj firj Kivelv fj TTia-TOTepois avra \6yois Kivuv tS>v vnodea-eav.) The writer lays down the general principle that we are bound to accept the assumptions and conclusions of the mathematician in the sphere of mathematics, unless very convincing arguments are brought against them.
^ ^^31-33. The first instance adduced by the writer to show that the theory of indivisible lines collides with ra iv toIs jxaOfjixaai nOeixeva.
We must suppose that it was customary in contemporary mathematics to define Ime as ' that which is between two points ', and straight line as 'that, the middle point of which is in the way of [blocks] both ends'. For the first definition, cf. perhaps Arist. Phys. 231^ 9, o-Tiyfiav 8' del to fi€Ta^v ypafxfir]. For the second definition, cf perhaps Plato, Panneft. 137 E, where that a-x^pn is said to be evBv "01! av to pia-ov ap^olv to\v icrxaTOiv eTTiTrpocrdfv ij '*.
^ At 970* 4 I accept Apelt's conjecture, del prjTop etrrai t6 TeTpdyavov for the MSS. 8iaip€T6v eorai to TCTpdyaPov.
This second instance (969^ 33-970* 4), in which the doctrine collides with mathematics, is a case partly of collision with the definitions of certain mathematical properties, partly of collision with certain demon- strated conclusions.
The writer complains that the doctrine of indivisible lines plays havoc (i) with the mathematical definition of * commensurate ' lines, and the mathematical distinctions which follow from it ; for since all lines what- ever consist of a whole number of these unit-lines, it follows that all lines are commensurate /n^/cei, and the mathematical distinction between surds and rational roots vanishes (969^ 33-970* 2) : and (ii) with the mathematical definition of ' rational ' squares, and the distinction between ' rational ' and * irrational ' squares which follows from it. For the indivisible lines
970*
DE LINEIS INSECABILIBUS
4 (iii) Again, since, in a rectangle, the line applied at right angles to the longer side determines the breadth of the figure : the rectangle, which is equal in area to the square on the indivisible line (v.g. on the line one foot long), will, if applied to a line double the indivisible line (v.g. to a line two feet long), have a breadth determined by a line shorter than the indivisible line : for its breadth will be less than the breadth of the square on the indivisible line.^
are all, qud infinitesimal, equal : hence all commensurate /Lii^/cet, and therefore also commensurate Surd/net (970* 2-4).
The point of the criticism is that the doctrine annihilates the mathe- matical conceptions of Commensurate and Incommensurate, Rational and Irrational.
The passage should be compared with Euclid, Eletn. X, deff. 2, 3 and 4 (see above, note on 968^14) : and with Plato, Theaet, 147D-148B. In the Theaetetus, Theaetetus and Socrates the Younger are represented as having generalized certain results of the mathematician Theodorus (their master), and having divided all numbers into two series, thus : —
Series i : Those numbers which, if regarded as the areas of rectangular
figures, are squares with whole numbers as their sides, e.g. 4, 9, 16, 25, &c.
The roots of these square numbers are what we should call ' rational ' :
or the sides of the squares are lines a-vfinerpoi iirjKci, viz. containing whole
numbers of the unit of length (the line one foot long).
Theaetetus and Socrates called the sides containing the squares in this series * fjLrjKT] '. Series 2 : Those numbers which, if regarded as the areas of rectangular figures with whole numbers as their sides, are oblongs ; or, if regarded as squares, have not whole numbers as their sides. To this series belong e.g. 3, 5, 6, 7, 8, &c. : and the sides containing these squares— e.g. ^3, VSf V^} &c.— were called by Theaetetus and his friend ' fiwa/xeiy ', i.e. dvvdfiei a-vfi- fxerpoi. (Cf. Theaet, 147 D J? tc rpiirovs /cat rj nevrenovs dvvdfxen are not p^Kci (TvpLficrpoi, rfj Trodiala. Id.f 148B a>s prjKiL pev ov arvppkrpovs cKfivais, Tois d* ennredois a dvvavTai.)
We should call the 'sides' of this series of squares * irrational square roots ' or ' surds '. ^ »4-8. In this passage I adopt Apelt's reading and interpretation throughout: v. Apelt, Aristotelis qtiae feruniur^ &c., -pro- legg. pp. xiv, XV.
If we suppose the ' indivisible line ' to be one foot long (cf. Arist. Met. 1052^^ 33— <"
Tox^ ypappals ;(pwi'rai o)S aTopat rrj Tro8iaia)f
then a rectangle, applied to a line two feet
B long, 7nust—'^ its area is to be equal to the
square on the indivisible line — have as its
than the indivisible line: which is absurd.
other side a line shorter
Let AB be the indivisible line, one foot long.
Let BE be the line, two
SECTION III 970*
(iv) Again, since any three given straight lines can be com- 8 bined to form a triangle, a triangle can also be formed by- combining three given indivisible lines. Such a triangle will be equilateral : but in every equilateral triangle the perpen- dicular dropped from the apex bisects the base. Hence, in the equilateral triangle whose sides are the indivisible lines, the ' indivisible ' base will be bisected by the perpendicular dropped from its apex.^
(v) Again, if the square can be constructed of Simples (i.e. n with indivisible lines as its sides), then let its diagonal be drawn, and a perpendicular dropped from one angle on to the diagonal. The square on the side (i.e. the original square constructed
feet long. Let CABD be the square on AB. If to the line BE there be applied a rectangular figure GFEB equal in area to CABD, FE or GB will be less than AB.
Though I accept Apelt's interpretation, there are one or two difficulties to which attention should be called, (i) napa^dWeiv is the technical term constantly used in Euclid (cf. e.g. Elem. I. 44, &c.) for * applying* a rectangle or a parallelogram to a given line : i.e. for constructing such a figure with a given line as one of its sides. But (so far as 1 know) it is always the figure which 'Trapa/SaXXtrai', not the side. Hence irapa- ^aWofxePTj here (970*5) is suspicious.
(2) Euclid constantly uses the technical expression 'ttXcxto? noul rfjv AB * to mean '' [a rectangle applied to such-and-such a given line] makes as its other side the line AB '. But, whatever may have been the original significance of the phrase, there is no implication in Euclid's usage that the side thus produced is shorter than the given line. So far as I have been able to discover, TrXaros -noiiiv in Euclid {a) always has the accusative (e.g. ^Tr\v AB') expressing the line resulting, and (d) does not mean ' determines the breadth *, but simply ' makes as its containing side (other than the given line) '. Cf. e.g. Euclid, Elem. X. 60, where the line thus produced is the longer of the two containing sides : and so often. But here (970* 5,^*7) the writer speaks of a line 'making the breadth' {to TrXtiroff Trotet), and the expression must be distinguished from the technical phrase in Euclid.
(3) In 970^6 Apelt reads tw ano rr\i droixov koi ttjs Trobiaias, to drro Ttjs dTOfxov means ' the square on the indivisible line'(cf. above, note on 968^ 14) : and we are to take the Kai as illustrative or explanatory. There is no serious difficulty here, though this introduction of the one-foot line is a little sudden. But the words in 1. 8 are very difficult. Apelt there reads eorai {yap} eXaTTov tov dno Ttjs drofxav, and the words ought to mean ' For it '—presumably, ' the breadth ' — ' will be less than the square on the indivisible line '. As this is nonsense, and as the alternative render- ing {' for it ', viz. l/ie rectangle, ' is less than the square ') gives a meaning irrelevant to the argument, we have to translate ' For the breadth of the rectangle Avill be less than that of the square '. But I cannot say that the Greek justifies this translation.
^ *8-ii. This argument presents no difficulty. Cf. Euclid, Elem. I. 10. ovv[(TTa<T6ai is the regular term in Euclid for * constructing' a figure.
970^
DE linp:is insecabilibus
with Simples as its sides) will be equal to the square on the
perpendicular together with the square on half the diagonal.
Hence the side of the square — i.e. the ' indivisible' line — will
not be the smallest line.^
14 Nor will the area, which is the square on the diagonal, be
double the square on the indivisible line. For (suppose it to
be so : then,) if from the diagonal a length equal to the side
of the original square be subtracted, the remaining portion of
the diagonal will be less than the ' simple' line. For if the
remaining portion of the diagonal were (not less than, but)
equal to the ' simple ' line, the square on the diagonal would
have been four times the original square.^
* ^11-14. I adopt Apelt's emendation Sm/ueVpou in 1. 12 for the MSS. dm fieaov, and in 1. 1 1 I read fl to Terpdycovov (ex) rcov dfiepcov (sc. (TvviaTaTni). The Latin translation by Rota has ' si quadratum ex qiiatuor insecabilibus lineis consistat',and LPW* omit Terpdyovov in a lacuna. Perhaps we should read d to reTpdyoiVov €K TCTTdpcov ajxepcov, or el €k TCTTapcov dfxepoiv TiTpdymvov. Another interpretation would be possible, if we retain the MSS. reading fl TO TfTpdymvov Ta>v dpepStv, but alter eXaxlcrTrj in 1. 14 tO e\a;^iOToi'. ' If the square belongs to the class of Simples, then ... [as above] . . . half the diagonal. Hence the "simple" square will not be the smallest square.' The argument would then be directed against the application of the theory of * simples ' to squares (cf. above, 968^14-16). The assumption of a least ' indivisible ' or * simple ' square collides with Euclid, E/em. I. 47. For, let A BCD be the ' simple ', or ' minimal ', square. Draw the dia- gonal BjD, and the perpendicular AE bisecting BD at E. Then, since AEB is a right angle, AB^ = AE'' + BE^ and therefore AE"" and BE"" are, each of them, smaller squares than the supposed smallest square, A BCD.
But the expression in 1. 12 (f) tov TCTpayoDvov nXevpd . . . ), and also the argument in 11. 14-17, seem decisive in favour of the interpretation which I have adopted in the text.
^ ^14-1 7. In 1. 17 1 read (with N and Apelt) « yap 'lar], TeTpairXda-iov av fypasj/fv r) tuip.cTpos. And after 8idp.fTpos I read a full stop.
Geometers have proved (i) that the square on the diagonal = twice the square within which the diagonal is taken : i.e. that BD^ = 2ABCD: and (ii) that if any line xy = twice any other line mn, xy^ — \mn^.
Hence, it follows that BD in the square ABCD is less than 2 AB : i. e. that, if from BD a portion DF = AB be subtracted, the remainder ^Z" is less than AB. If, therefore, AB is an * indivisible ' line, either BD"^ will not be equal to 2 AB ^ (but = at least 4AB^), or BD will contain FD { = AB) + BE (a line less than the * indivisible ' line) : the first alternative conflicts with an established geo- metrical conclusion, and the second alternative is absurd.
SECTION III 97o»
And one might collect other similar absurdities to which the doctrine leads ; for indeed it conflicts with practically every- thing in mathematics.'
(B) Then again (the following arguments support our criti- cism of the doctrine) : — ^
(i) The Simple admits of only one mode of conjunction, but 19 a line admits of two : for one line may be conjoined to another either by contact along the whole length of both lines, or by contact at either of its opposite terminal points.^
(ii) Further, the addition of a line will not (on the theory) make the whole line any longer than the original line to which the addition was made : for Simples will not, by being added together, produce an increased total magnitude.*
(iii) Further, every continuous quantum admits more divi- 23 sions than one, and therefore no continuous quantum can be formed out of two Simples. And since every line (other than the indivisible line) is admittedly continuous, there can be no indivisible line: (for if there were, a continuous quantum — viz. the line formed by the conjunction of two indivisible lines — would be formed out of two Simples.)^
In 1. 16 d<l)aipedevTos yap rov taov, we should presumably understand
fJLTjKOVS. )
^ ^17. The MSS. read aXXn 8* uv ns Kal erepa kt'K. Apelt conjectures (^ (iXoya d* av kt\. There should, of course, be a full stop between bidfierpos '
and aWa (or aXoya).
"^ ^19. This begins a second series of arguments (in support of the writer's rejection of indivisible lines). noKiv here corresponds to irpStrov H€v . . . (969^ 29), which introduced the series of arguments just con- cluded.
^ ^19-21. What is * simple ' or ' without parts ' can be conjoined with anything else only in one fashion. But a line can be («)laid alongside of another line, or (d) conjoined with it, end to end. (Cf. //e Cae/o, 299^ 25).
The words in ^21 Kara to nepns e^ evavrlas {evuvriov LP) are obscure. I take them to mean * at either of its
contrary terminal points '. The 7node a B
of (Tvva^is is the same whether the line ^ y x y
xy be conjoined with the line AB sit A or at B, and at x or at y. '
* a2i-23. Apelt conjectures (from Pachymeres) en ypapfifj {ypafipfi} TrpoaTedela-a ...
The addition of ypnixixfj makes the Greek easier, but does not seem absolutely necessary.
5 323-26. I adopt Apelt's reading en (ei) ck 8vo7v dp,cpoiv firjbeu yivtrai {yivfo-Oai MSS.), and also his punctuation, but not his interpretation.
I have paraphrased freely, so as to bring out the argument as I under-
C 2
970^ DE LINEIS INSECABILIBUS
j6 (iv) Further, if every line (other than the indivisible line) can be divided both into equal and into unequal parts — every line, even if it consist of three or any odd number of indivisible lines — it will follow that the ' indivisible ' line is divisible.^
stand it. The writer assumes {anaa-a Be ypafifxr] irapd ttjv arojiov a-vvcx^js) that even the advocates of indivisible lines admit that all o^/ier lines are continuous : and argues that a line compounded of two indivisible lines would, on their admission, have to be continuous, but could not be so on the principle that every continuum admits more than one division.
1 a26-28. The MSS. read crt ci cmatra ypanfirj napa [nepl LNP, Om. Z*] T^s drofMov Kai laa [koi els laa L] koI civiara diaipelrai kol firj (k rpiav drofioiv Koi oXcop nepiTTOiu atcrr' ddialperos rj aTOfxos.
I accept Apelt's reading (which is partly based on Hayduck's conjec- tures) €Tt (I anaaa ypafipr] napa rqp arofiov Kal (Is Icra koi aviaa SiaipelTai, kov fl (K rpiSiv Kai oXcos nepiTTOiP, earTai diaiperq fj arofios.
The writer is assuming, in the present series of arguments (970* 19-33), that the advocates of indivisible lines accept certain common mathemati- cal assumptions as applying to the composite (non-indivisible) lines : and shows that their application is inconsistent with the * indivisibility ' of the unit-lines.
The assumption here stated is airaaa ypapp.f) koI els taa *cat avKra diaipei- Tai. This formula is constandy used by Euclid (cf. e.g. Eiem. II. 5 and 9) to mean bisection and simultaneous division into two unequal parts. If we so understand it here, the argument is plain : but then 1. 33 {orav 17 Ik rSav dpTiwv els aviaa diaiprJTai) is unintelligible.
It seems best, therefore, to interpret ' into any nufnder of equal, and a?ty number of unequal parts '. And there is reason for thinking that ' division into unequal parts ', as here contemplated, involved a process of progres- sive bisection. (Cf. e.g. Alexander's Commentary on Arist. De Sensu^ 445^ 27 : and G. R. T. Ross, Aristotle : De Sensu and De Memoria,
pp. 199-200.) If, e.g., the line AB
^ O c B ^^^s ^^ ^^ divided into \ and |, the
method would be to bisect AB at C\ and again to bisect ^C at D. AD would then be I, and DB |, of ^^. It would not be possible by this method to divide AB into parts repre-
\
/ ^
Vt
«
\
/
/
/ /
c — %y€,? f
ffi
sented by fractions whose denominators were other than powers of 2 ; but it would be possible to exhibit such fractions on the line AB. Thus, e.g..
SECTION III 970*
And the same will result if every line admits of bisection : 29 for then every line consisting of an odd number of indivisible lines will admit of bisection, and this will involve the division of the * indivisible ' line.^
it would not be possible to divide AB into f and f , nor into f and ^. But by triply bisecting AB, and eliminating ^th, the remainder AI could be divided into AG = f and GI = f : whilst, by eliminating fth, the remain- der v4F could be divided into A/I = % and HF = \.
There is no evidence in this passage that the writer knew of the follow- ing method for dividing any given line into any number of parts :— Let it be required to divide AB into (e. g.) three equal parts. From B draw BC
= AB, produce BC to D, making CD = AB: and produce BD to E, making DE = AB. Join EA ; and from D and C draw DE and CG, each parallel to EA, to the points E and G on AB. AE, EG, and GB will then be, each of them, ^rd of AB.
If we assume that the writer was unaware of this latter method, it is obvious {a) that no line consisting of an odd number of unit-lines could be ' divided into unequal parts ', for the first bisection would divide the middle unit-line : and {b) that there would be a limit to the ' division into unequal parts ' of lines consisting of an even number of unit-lines, since no such line could be progressively bisected ad libitwn without dividing the unit-line (cf. 970* 33).
^ *29, 30. Mathematicians further assume that every line can be bisected. If the advocates of * indivisible ' lines accept this assumption, it will apply to lines compounded of an odd number of unit-lines (Tracra yap f) cK t5>p TrepiTTotv, sc. Si'^a refiverai) : but they cannot be bisected unless the middle ' indivisible ' line is divided.
97o^ DE LINEIS INSECABILIBUS
And if not every line, but only lines consisting of an even number of units admit of bisection : still, even so, the ' indivi- sible ' line will be divided, when the line consisting of an even number of units is divided into unequal parts (by progressive bisection).^ 33 (C) Again/'^ (the following arguments must be considered against the doctrine) : — 970^ (i) If a body has been set in motion and takes a certain time to traverse a certain stretch, and half that time to traverse half that stretch, it will traverse less than half the stretch in less than half the time.^ Hence if* the stretch be a length con- sisting of an odd number of indivisible unit-lines, we shall here again find ^ the bisection of the ' indivisible ' lines, since the 5 body will traverse half the stretch in the half time : for the time and the line will be correspondingly divided.^
So that none of the composite lines will admit of division both into equal and into unequal parts, nor will they admit of
^ '^So-SS- In the above interpretation I have omitted altogether the words rr]v 8e hixa diaipovfxeur]v Koi oaa dvuarov renveiv. These words as they Stand will not translate. If we read /cat els avia-a in place of koi oa-a, the meaning is plain enough : but the words are then not required for the argument.
Hayduck, and after him Apelt, conjectures koi oa-aovv 'and if it is possible to divide (i. e. to bisect) the line which is being bisected (viz. the line with an even number of units) as many times as you please *. But, if my interpretation of bunpeais els avia-a is right, these words are not required. Whilst, if my interpretation is wrong, I do not see how a valid argument is to be extracted from the passage. Apelt (cf. his Prolegg. p. xviii, note, p. xix : and his German translation of the passage) interprets avicra as equivalent to Trfpirra, for which I can discover no justification.
^ ^"hZ' TraXip 61 ktK. This TrdXtp answers to -nakiv tov fxev dfiepovs (970'* 19), and marks the beginning of a new group of arguments.
^ The protasis extends to KivndT}<T€Tai, and the apodosis is koI ev tw fXiiTTovi . . . T][xiaeinv. We should therefore place a comma after KivrjOrj- (TCTai {()7d^ 2).
^ ^3. I adopt Apelt's conjecture et pih (ex) nepiTTHv.
^ ^3. The MSS. read dvaipedqaeTai (Z^ fort. dvepeO^^aerai). Apelt con- jectures av ivpedfja-erai, but the position of the av is impossible. I read duevpedrjorcTai (' redibit ', Rota).
^ ^5, 6. Since the time is bisected, the stretch — i.e. the line, supposed in this case to consist of an odd number of units — will be bisected too.
After these words there is, I think, a lacuna. For nothing is said as to the case in which the stretch consists of an even number of units :■— i.e. there is no clause to answer to et jacv ck ncpiTToav in 970^ 3. And no use is made of the thesis established in 970^2 {miiv t(o iXdrTovi . . . Trjvfjpia-ciav), which was probably intended to be applied in proving the divisibility of the unit-line, even when the stretch consisted of an even number of units.
SECTION III 970^
division corresponding to the division of the times, if there are to be ' indivisible ' lines.^ And yet (as we said) the truth is, that the same argument, which leads to the view that lines consist of Simples, leads by logical necessity to the view that all these things (composite times, e.g., as well as composite lines) consist of Simples.^
(ii) Further, every line which is not infinite has two terminal 10 points : for line is defined by these. Now, the ' indivisible ' line is not infinite, and will therefore have a terminal point. Hence it is divisible : for the terminal point and that which it terminates are different from one another. Otherwise there will be a third kind of line, which is neither finite nor infinite.^
(iii) Further, there will not be a point contained in every 14 line. For there will be no point contained in the indivisible line ; since, if it contains one point only, a line will be a point, whilst if it contains more than one point it will be divisible. And if* there is no point in the indivisible line, neither will there be a point in any line at all : for all the other lines are made up out of the indivisible lines.^ 18
^ ^7, 8. I read with Hayduck ouS* 6fxoi<os rois ;(poi'ots TfirjBrjaovTai, el [MSS. ovk] eaovrm . . . The whole sentence is intelligible only if we assume that something has dropped out between rfXTiBrjaeTai and axrve in 1. 6 : see the preceding note.
^ "8. TO. de Tov avTOv \6yov eari, KaBdnep €\i)(6r], to navra Tavra noieiv e^ dfiepcov.
The reference is to 969* 29, 30. For ra Se we should presumably read TO fie. By ndvTa TavTa we must understand prmiarily firjKrj and XP^^^*- • but no doubt the statement is intended to apply to all composite quanta.
^ ^10-14. In ^12 I read (with Bekker) nWo for the MSS. dWov. Every line, unless it be infinite, has two ends or limits, viz. its terminal points. The indivisible line, therefore, since it is not infinite, has two limits. But, if it has even one limit, it is divisible, viz. into {a) the limit, and {b) the limited. The only escape from this dilemma {^either infinite or limited and so divisible ') would be to say that the ' indivisible lines ' constitute a third class of line, neither finite nor infinite.
* ^17. (I nev ovv . . . What is the exact force of ' filv ovv ' here .'' Does it mean ' And, what is more, if ? Or * And if it be conceded that * ?
^ ^14-18. In 11. 15-16 I read (with Apelt) d nev yap p.ia p.6vr) evvnap^ei, ypafxp-rj toTai aTiyprj for the MSS. el fjtev yap pia [paKiaTa LPZ**'] povj} VTrdp^ei ypappr], eira aTiypr).
The writer sets out to show that the geometrical principle, that * in every line there is contained a point ', will not hold of the ' indivisible ' line. For if it contains but one point, it will be that point, i. e. a line will be a point : whilst if it contains more than one, it will be divisible. He then shows that it follows that this geometrical principle does not hold of any line, since all lines are (on the theory) either indivisible lines or com-
97o^ DE LINEIS INSECABILIBUS
Moreover, if there are points in the indivisible line, there will either be nothing between the points, or a line. But if there is a line between them, and if all lines contain more points than one, the unit-line will not be indivisible.^
21 (iv) Again, it will not be possible to construct a square on every line. For a square will always possess length and breadth, and will therefore be divisible, since each of its dimensions — its length and its breadth — is a determinate something. But if the square is divisible, then so will be the line on which it is constructed.^
23 (v) Again, the limit of the line will be a line and not a point.^ For it is the ultimate thing which is a limit, and it is the * indi- visible line ' which is ultimate.* For if the ultimate thing be 'point', then the limit to the indivisible line will be a point, and one line will be longer than another by a point.^ But if it be urged that the limiting point is contained within the
posites of these. For the geometrical principle cf. Arist. Post. Anal. 73* 31 Kai ft eV Trdtrj; ypafXfijj ariy^rj . . .
* ^18-20. I interpret this as a further argument to prove that there cannot be two (or more) points in the indivisible line. For suppose there are two points in it. Then either there is nothing between them, and then they collapse into an indistinguishable unity: or there is a line separating them. But then this line will itself contain two or more points, between which there must be another line, and so on in mjinitum : hence the original unit-line will not be * indivisible ' if it contains two (or more) points.
2 ^21-23. This argument is very obscure, and perhaps the text is wrong. It is a principle of geometry that a square can be constructed on any given ^ g line : but it does not follow, because the length {AB)
of the square ABCD is distinguishable from its breadth {A C), and because therefore the square is divisible into length and breadth, that AB or AC are themselves divisible gttd lines.
The Greek cVfi to /xcV, to de ti seems suspicious, but I have no remedy to propose. Cf., however, the argument at 970^ 30 ff. A square, if divided, must
C O be divided ' at a line ' : i.e. its division must involve
the division of its breadth or length. But this is impossible if its sides (and therefore all lines within it which are parallel to them) are * indivisible ' lines.
* In ^24 I read with Apelt (after Hayduck) ypafi^ir] €<TTai, dXX' ov a-Tiyfirj for the MSS. arTiyfir) earai [eiXTiv N], dXX' ov ypanfirj. N's eaTiv is a trans- parent, but futile, attempt to make sense of the traditional reading.
* In^25 1 accept Bussemaker's conjecture t6 ca-xaTov, {eaxarov) dt fj aTOfxos. ^ In ^25 I retain the MSS. reading el yap aTiyfiij [sc. to ecrxaTov], to irepns
Tjj drofico eorai <rTiyp,r]. Apelt's conjecture, el yap (TTiyp-q to trepas, {nepas} T.V drofjuo eo-Tai a-Tiyixrjf though it would be convenient, is not necessary.
SECTION III 970^
indivisible line, on the ground that two lines united so as to form a continuous line have one and the same limit at their juncture, then the simple line (i.e. the line without parts) will after all have a limit belonging to it.^ 28
And, indeed, how will a point differ at all from a line on their theory ? For the indivisible line will possess nothing characteristic to distinguish it from the point, except the name.^
(vi) Again, if there be indivisible lines, there must, by parity .^o of reasoning, be indivisible planes and solids too.^ For the being of an indivisible unit in one dimension will carry with it the being of indivisibles in the remaining dimensions too,^ since it is at a plane that a solid is divided, and at a line that a plane is divided. But there is no indivisible solid : for a solid contains depth and breadth. Hence neither can there be an 97^^ indivisible line.^ For a solid is divisible at a plane, and a plane is divisible at a line.^
1 ^23-28. TO ea-xarop is the ultimate (or most elementary) thing in the spatial sphere: the not-further- reducible element of extended quanta. On the hypothesis of indivisible lines (the writer urges) this ultimate element of extension is the unit-line, and not the point. If it were the point, then eMer (a) the point limits the indivisible line ad extra, in which case the addition of a point would increase the length of a line : or {b) the point, which limits the indivisible line, is internal to it : but then the internal limiting point will be a distinguishable part of it, i.e. of that which is ex hypothesi without parts (cf. 970^ 12, 13).
In 11. 27, 28 the words hia to tovto Trepas tS>v <rvv€)(ovaciiP ypafifiStv (sc. eivai) indicate the grounds on which (d) might be maintained. If the line CD be joined to the line AB, so as g
to make a continuous line A£>, B A ——————— D
and C become one and the same
point, the end of ^^ and the beginning of CD (cf. Arist. Phys, 272*
10-13).
^ ^'29, 30. oXws- re [read hk with N] tI dioiaei (TTiyjxr) ypafxfirjs ; The writer has just shown that the theory leads to the difficulty that a line must be terminated by a line and not by a point. From this special difficulty he now passes to the general difficulty that, on the theory, there can be no difference between ' point ' and ' line ', except in name.
^ ^31. The MSS. read en [en el N] ofioicos /xevei errinedov *cat (rai/Lia eaTiv arofwv. For fievei Hayduck proposed fjuJKd, and Apelt fxh mi. I accept Apelt's conjecture, and agree with Hayduck in reading fo-rai for iaTiv. In ^33 the MSS. read trai/xa ovk co-rai [earlp NZ*] d8iaip€Tov . . . : but we must follow the edt'tzo princeps and insert hk after atiipja. This Sc will then correspond to the [lAp in ^31. I agree with Hayduck and Apelt in reading NZ*'s itTTiv in ^33 in place of earat.
* ^31,32. Literally, * For if one is indivisible, all the others will follow suit.*
^ *i. I read with Apelt ovd' {ap^ ap ypafifxrj cii; . . .
^ ^^30-97 1* 3. If there are simple lines, there must be simple planes— viz.
971^ DE LINEIS INSECABILIBUS
3 But since the arguments by which they endeavour to con- vince us are weak and false, and since the opinions (which they are trying to establish) conflict with all the most convincing arguments, it is clear that there can be no indivisible line.^
§ 4. And it is further clear from the above considerations that a line can no more be composed of points than of indivi- sible lines. For the same arguments, or most of them, will apply equally against both views.
7 For (i) it will necessarily follow that the point is divided, when the line composed of an odd number of points is divided into equal parts, or when the line composed of an even number of points is divided into unequal parts.^
the planes bounded by those lines— and if there are simple planes there niust be simple solids, viz. the solids contained by those planes. For to divide a solid is to divide it at a plane, and thus to divide all the planes at right angles to the plane of division. And to divide a plane (cf. above, 97Q^ 21-23) is to divide it at a line, and thus to divide all the lines at right angles to the line of division. Hence if every solid, however minute, is ^ Q divisible, every plane must be divisible too : and if
every plane, however small, is divisible, every line must be^divisible too.
This appears to be the argument ; but the reason given (971* i) for the divisibility of every solid is obscure, in the same way as the reason given for the divisibility of every square (970^ 23) was not convincing. And could not the advocates of * indivisible lines ' have insisted that a plane figure, though divisible, might yet have as one of its con- taining sides an ' indivisible line ' ? The oblong ABCD, e.g., might be divisible along its length AB, " ^ and yet indivisible in respect to its breadth AD: i. e.
AD might be an * indivisible line '.
* ^3-5. This sums up the case against the indivisible lines. We have seen in § 2 that the arguments advanced in support of the theory are weak and false : and we have seen in § 3 that the tenets of the theory collide with the principles and conclusions of mathematics.
The text in these lines is not very satisfactory. We should expect a somewhat stronger particle than 8e in *3 to introduce a summing-up of this kind : but it is difficult to make a convincing emendation. The T€ {01 re \6yoi) is apparently answered by dt in 1. 4 (ivaurlai 8e 86$ai), which is omitted by all the MSS. except N. Perhaps the grammatical structure is ol X0701 . . . dadivf'is T€ Koi ylrevdels elai ? See Bonitz, Index, 749^ 44 ff.
All the MSS. in 1. 4 read naam except P, which has naai. Neither reading is entirely satisfactory. There seems no point in naaai, and naai is not strictly true— or at least has not been shown to be true.
Tois ia-xvovai [sc. \6yois] it^jos tt'kttiv — * the arguments strong to produce conviction ' are presumably the mathematical arguments : cf. e.g. 969^ 30.
* ^7-9. I adopt Hayduck's conjecture j) (17) (< nepnTaip and rj <^) e^ apTiatv ...
SECTION IV 971*
And (ii) it will follow that the part of a line is not a line, nor the part of a plane a plane.^
Further (iii) it will follow that one line is longer than another 10 by a point ^ : for it is by its constituent elements that one line will exceed another. But that it is impossible for one line to be longer than another by a point, is clear both from what is proved in mathematics and from the following argument. 1 For, if it were possible, the absurd consequence would result ^ that the moving body would take a time to traverse the point.^ For, as it traverses the equal line in an equal time, it will traverse the longer line in a greater time : and that by which the greater time exceeds the equal time is itself a time.
Perhaps, however, we are to suppose that just as a line con- 16 sists of points, so also time consists of * nows ', and both theses belong to the same way of thinking. (Let us then examine the doctrine that a line, or generally continua^ like times and lengths, consist of discrete elements.)*
In 1. 9 TO. avia-a is Strange : Z* omits ra.
The reference is to the obscure argument at 970* 26-33.
^ *9, 10. If a line is made up of points, a plane on the same principle will be made up of lines : and the ' parts ' of a line will be its ' points ', and of a plane its ' lines '.
^ *lo, II. The MSS. read koX ypaixfxr} 8e ypafinrjs (TTiynfj [(TTiyfif) W% ariyn^s N] dvai fni^av'
I read, with Hayduck, koX ypafifiqv 8e ypa/x/M^? a-Tiynfj eJvai ficiCo*'
' *I3. Trjv aTiyfirjv, i.e. tAe point, by which the longer line exceeds the shorter. I accept Hayduck's diUvai for the MSS. dfj eluat.
* The writer is led off, by a possible rejoinder, to consider the view that time consists of 'nows'. But in the series of arguments which follows, the first argument alone directly mentions ' time ' and ' nows ' : and though some of the subsequent arguments would apply to 'time ' as well as to the line, many of them apply specially and only to lines. Hence I interpret 971^*3 and 4 as a corollary^ and not as a summary: and I regard the whole of \ Ac (971*6-972^13) as a connected series of arguments to show that a line cannot consist of points. The order of the writer's thought is, I think, as follows : —
(i) 971*6-16. Statement of the arguments which are fatal both to the doctrine that a line consists of indivisible lines, and to the doctrine that it consists of points : and statement of a new difficulty against the latter doctrine. This difficulty involves the conception of Time, and might be met by the rejoinder that Time, like Length, though continuous, consists of discretes. (2) 97 1» 17-972* 13. A group of arguments to show that a line cannot consist of points, the view that Time consists of Nows being incidentally refuted. This group of arguments is based on a disjunction, thus : — The points cannot be united to form the line either (a) by awexetn (971* 17-20), or (d) by avvOea-is (97 1* 20-26), or (e) by &(f>r) (97 1» 26-^26), or [d) by to €(f)€^fjs (971^ 26-972* 6). '
971* DE LINEIS INSECABILIBUS
17 (a) Since, then, the Now is a beginning and end of a ^ time, and the Point a beginning and end of a line ; and since the beginning of anything is not ' continuous ' with its end, but they have an interval between them ; it follows that neither Nows nor Points can be continuous with one another.^
20 (b) Again, a line ^ is a magnitude : but the ' composition ' of points constitutes no magnitude, because several points put together occupy no more space than one. For when one line is superimposed on another and coincides'^ with it, the breadth is in no wise increased. And since points too are contained in the line thus superimposed, it follows that neither would points, by being superimposed on points, occupy more space. Hence points would not constitute a magnitude by composition.^
Of these four alternatives (TvvOearis is used by Aristotle as the general term to express any kind of combination of a manifold : cf. e. g. Top. Z 13, 150^ 22, Z 14, 151* 20-32. Here, however, as we shall see, the writer appears to use it to express one special kind of combination. The remaining alternatives are treated by Aristotle as exhausting the ways in which points might be supposed to cohere to form a line : cf. Arist. Pkys. 23i»i8fif. Aristotle's definitions {Phys. I.e.), which the writer here assumes, are ' frwiyj] \ikv lav ra ea-xnra ev, anTofxeva 5* Sti/ afia, ecfie^fjs 6* a)V fxrfdcu fxera^i) (Tvyyeves '.
^ *i8. Tov xpoi'oi^j i'Cf any given period of time.
^ ^17-20. Two things are called * continuous' when the end of one is identical with the beginning of the other. But the Nows and the Points are themselves Ends and Beginnings, or Extremes (eo-xara), and cannot therefore be * continuous ' with one another.
^ ^21. f] fih ypafifiT] 'the line', i.e. any and every line: cf. 971* 18, TOV xpovov.
* ^23. For this use oii^apn6((tv cf. e.g. Euclid, Elem. I. 4, '*€<^ap^oorfi Kcu TO B crT}fi€iov eVi to E . . ."
^ *2o-26. In this argument the writer seems to be excluding a view that point is applied to point so as to ' compound ' a line. Line is length without breadth : and if line be applied to line, the two coincide, fall on one another, and do not produce a surface, i.e. do not * increase the breadth ' of the first line. So point is position without magnitude, and no application (composition or addition) of point to point can produce magnitude — i.e. length. If the line AB be applied to the line CD, the points in AB will coincide with the
points in CD : and as the line CD is A y ____B
no 'broader' than it was before, _^___^
neither will any point x in CD be- C «, D
come a length by ' composition ' with
the corresponding pointy in AB. There is some difficulty in the text. In 971* 22 the MSS. read dia to fxrjd' eVi TrXeiw tottoi' ex""* ShouldgWe perhaps read dta to /xj/S' 6ti TrXf/w tottov KUTex'^iv ? In 1. 24 I ^ retain the MSS. reading iv de tt} ypa^ifx^ . . . (Apelt's emendation « di tj} ypainirj . . . does not suit the movement of the argument.) But I read
SECTION IV 971*
(c) Again, whenever one thing is ' contiguous ' with another, 26 the contact is either whole-with-whole, or part-with-part, or whole-with-part. But the point is without parts. Hence the contact of point with point must be a contact whole- with-whole.^
But if one thing is in contact with another whole-with-whole, the two things must be one. For if either of them is anything in any respect in which the other is not, they would not be in contact whole-with-whole.2
But if the Simples (when in contact) are (not ' one ', but) 30 * coincident ', then a plurality occupies the same place which was formerly occupied by one : for if two things are coincident and neither admits of being extended beyond the coincidence, just so far the place occupied by both is the same. And since 97^^ the Simple has no dimension, it follows that a continuous magnitude cannot be composed of Simples. Hence neither can a line consist of Points nor a time of Nows.^
in 1. 25 oi»5' av (apy al aTiyfial . . ., and alter the punctuation, so that the whole passage runs as follows : —
. . . fiei^op TO ttXcitos' iv de Trj ypafifxfj koI ariyfial ivimdp\ova iv' ou8' av (np) al (TTiypLoi ttXcio) Karexotep totvov, cotTTe ovk av rroioUv fieyfSos.
' In »27, 28 I read with Apelt (after Hayduck) rj 8e oriy/xy afiepfjs, oXa)s (a;') anroiTO.
The principle that all contact must be whole-with-whole, or part- with-part, or whole-with-part, is enunciated by Aristotle {Phys. 231^2), and applied similarly to ddiaipfra and specially to points.
^ ^29. The MSS. read el ydp n [ns NZ*] eVrii/ rj Odrepov fifj iariv . . .: I read 17 ddrepov (cf. the Latin transl. ' si quid remanet quod alteri non coniungatur ').
Apelt conjectures el ydp dls (or bv) ecrriv . . . ' si totum bis est vel non simul alterum complectitur . . .'
^ ^26-^4. The outline of the argument is as follows : — The contact of Points, gud Simples, must be whole-with-whole. Now two things are ' contiguous ' when their extremities are dp.a, ' coincident ' or ' together '. But since Simples have no parts— no extremities in distinction from the rest of themselves— the contact of Simples must mean absolute unity. If this be denied, and it be maintained that the ' contiguous ' Simples are * coincident ', but remain ' two ' : it will follow that two or more Simples can be ' coincident ' without taking up more place than one Simple, and therefore (since one Simple has no dimension, i.e. no inner extension) no continuous magnitude can be composed of Simples. And a corollary of this is, that a line cannot consist of points, nor a time of ' nows '.
In 971^ I I read, with LPW*Z*, eVeKTao-tr', Kara ravTa 6 avros ktX. Apelt's conjecture (eneKTaoiv koO' iavrd, 6 avros . . .) is tempting, but un- necessary.
In 971^ 2 5tao-Ta(rty= dimension, cf. Bonitz, Index, 189* soflf.
971
DE LINEIS INSECABILIBUS
4 (d) Further, if the h'ne consists of points, point will be in contact with point. If, then, from K there be drawn the lines AB and CD, the point B in the line A[B)K and the point C in the line K{C)D will both be in contact with K} So that the points B and C will also be in contact with one another : for the Simple, when in contact with the Simple, is in contact whole-with-whole. So that the points will occupy the same place as K, and, qtid in contact with K, will be in the same place with one another. But if they are in the same place
lo with one another, they must also be in contact with one another : for things which are in the same ' continent ' place must be in contact.^ But, if this is so, one straight line will touch another
* ''4-6. The writer assumes for the prese7it that, if a line is made up of points, the points within the line are in contact with one another. Having laid down this assumption, he then proceeds (fa»' olv — : oZv is omitted by LP, but is required) to suppose that from the point K two lines, each consisting of points, are drawn. He calls these lines ' AB' and ' CZ>' ; but it is clear, from what follows, that the points B and C are the terminal points of the lines contiguous to A', i. e. that A and D are ^ the end-points furthest removed from K. This is directed to prove that, since B and C are in contact with A', they are also in contact with one another. The text is corrupt, and I have ventured to read and punctuate as follows : —
wore KCLi aKKr]Koiv [so Apelt for the MSS. aXXcoi/, aXXo), aXXw Ttw. Hayduck conjectured aXXiyXcoi']. t6 yap dfiepes too dfiepovs o\ov oXou e(f)dnTeTai' ayare top avTov ecfie^ei tottov rat K, koI tov K anrofifvai ai trrfy/iai €v TO) avTa TOTTO) aXX^Xai?. el 8' eV to) avTco, Koi airTovTai' ra yap ev tw avrm TOTTO) ovTa TrpMTO) [so Hayduck for the MSS. 7rp5ra or 7rpa>Tov] anTfo-dai dpayKaiov.
For the meaning of Trptoro), cf. e.g. P/iys. 209^326"., Kal ronos o fxev KOivoSf iv 0) CLTvavTa to, a^fiard eaTiv, 6 S' idios iv o) npoyTW, . . . The * proper ' or * primary ' place of a thing is further explained as that which contains precisely the thing and nothing more, i. e. the continent boundary of the thing. Cf. also P/iys. 226^ 21-23.
The argument moves thus : ' B and C are in contact with IC. But B and C are points, i.e. Simples. And contact of Simples is contact whole-with-whole, i.e. complete coincidence. Hence the "continent place " of B is identical with that of A', and the " continent place " of AT is identical with that of C. And therefore the " continent place " of B is identical with that of C. But this means that -5 is in contact with C*
In 971^ 8 the MSS. read . . . 60c^« tottov tov K, koi aTTToixei/m o-riyfxal . . . Apelt conjectures i^e^ei tottov (r« K' eaovTai ovv Kal at) tov K aTTTofifvai (TTiypni kt^. This involves more change than the reading which I propose: and, after all, it is not satisfactory. For the writer shows that B and C, qua points in contact with a third point, K — i. e. qua
W-
SECTION IV
971
straight line in two points. For the point (B) in the line AK touches both the point KC and another (viz. the point con- tiguous to C in the line K{C)D). Hence the line AK touches the line CD in more points than one.^
And the same argument would apply not only in the case supposed, where two lines were in contact with one another at the point K^ but also if there had been any number of lines touching one another at K.^
in contact with K whole-with-vvhole — must have one and the same ' continent place ' as K, and therefore as one another : and therefore must be in contact with one another. The nerve of the argument is contained in the words ' and /^e points^ because in contact with K"* : but Apelt's reading could only be translated ' Therefore the points which are in contact with K will also be in the same place as one another *. (Apelt's note on 1. 9 « b' iv rcS avra ... * scribendum potius videtur yap ', shows 'that he has failed to follow the writer's argument.)
^ 'm-14. The writer, having proved that the terminal points B and C are in contact at Ky shows that the two straight lines £A and CD will be in contact at more than one point— v. g. at x, since C is in contact with x and B with C.
At 1. II I adopt Hayduck's el 8' ovtcos for the MSS. (W ovTcos, and I read a full stop before these words.
At ^12, 13 I read Ka\ t^s KF (for the MSS. Koi rfi Kr) Koi ire pas . . . Apelt follows Hay- duck in reading *cai (ttjs eV) tjj KT. But ' Kr * is the <TTiyp,T] Kr, no;f the ypanixrj. If the writer had meant /he line, he would have written KA or TA as in 1. 6 or in 1. 13 (tj}s TA).
Finally, in 1. 13 I read (with Hayduck and Apelt) coo-re 17 AK in place of the MSS. Qjcrre et 6K or r\ (k.
^ ^14, 15. The MSS. read Ka\ d pf} 8l [Be N] aXXj/Xwi/, aKX* oTraxrovv rjy^aro ypapprjs [orroa-ovv rj'^aTo ypappr)
Z*]. I have adopted Apelt's
conjecture koI et pfj 8v dWrjXoyv, riW oTTOa-aiovv rj-^avro ypappai.
If this is right, we must suppose a number of lines, e.g. AB, CD, EF, GH, IJ, all drawn from K. The points A, C, E, G, /, qud all in contact with K, are all in contact with one another : and also severally in contact with the points X, y, z, p, q. Hence the lines AB, CD, EF, GH, IJ will be in contact with one another at more points than one.
971^ DE LINEIS INSECABILIBUS
15 (e) Further, if a line consist of points in contact with one another, the circumference of a circle will touch the tangent at more points than one. For both the point on the circumference and the point in the tangent touch the point of junction and also touch one another.^ But since this is not possible, neither is it possible for point to touch point. And if point cannot touch point, neither can the line consist of points : for if it did, they would necessarily be in contact.-^
20 (f) Moreover, how — on the supposition that the line consists of points — will there any longer be straight and curved lines ? For the conjunction of the points in the straight line will not differ in any way from their conjunction in the curved line. For the contact of Simple with Simple is contact whole-with- whole, and Simples admit no other mode of contact. Since, then, the straight and curved lines are different, but the con- junction of points is invariably the same, clearly a line will not be curved or straight because of the conjunction : hence neither will a line consist of points.^
^ ^15-18. Let the circumference of the circle DEA, and let the tangent CB, both consist of points. The point of juncture, x, will be in contact with Q the point B of the tangent CB{x)^ and also with the point A of the circumference DEA{x) : hence the point A will also be in contact with the point B. And the tangent CB{x) will touch the circumfer- ence DEA{x), at .<4, at x, and at B.
^ ^18-20. In 1. 20 I read, with Hay- duck, oirS' etVai Tr]v ypafxfx^v <TTiyna>v [MSS. a-Tiyfiqv. Perhaps we ought to read €k <TTtyixa>v]' ovTco [MSS. and Apelt oide] yap anreadai duayKolou.
Apelt defends ovde ' si linea ex punctis constaret, necessario a contactu exclude- retur (quod tamen fieri nequit) '. And, in his German translation, he interprets * Denn sie (die Linie) ware dann notwen- dig von der Beriihrung ausgeschlossen '.
But the Greek cannot mean this : nor, if it could, would there be any valid
argument in the words.
^ ^20-26. In 1. 24 I read (with Apelt and Hayduck) aXKcos a-nxfaOai for
the MSS. okas [oTTo)? W^] aiTTfoBaL.
11. 24-26 are difficult. I take the writer to mean : ' The theory rnight
attempt to distinguish Straight from Curved, on the ground that point is
attached to point differently in these different types of line. But points
are Simples, and therefore point can be attached to point in one way only.
Hence we cannot derive the different characters of the straight and curved
SECTION IV 971^
(g) Further, the points (of which the line consists) must 26 either touch or not touch one another. Now if ' the next ' in a series must touch the preceding term, the same arguments, which were advanced above, will apply : but if there can be ' a next' without its being in contact (with its predecessor or successor), yet by 'the continuous' we mean nothing but a composite whose constituents are in contact. So that the points forming the line must be in contact, in so far as the line must be continuous, even though we suppose the points to be a * series '.^
(h) f ert et cltottov ortyjur/ eTrl ariyixrj'i [eTrtfTTTJ/xr; Z**], tv f\ 97* [^ PZ*] ypaynxi] kolI cttI (TTiyySi, [ypaju/x?/ koX iinarTrJix-qs NW'*', fTrto-TTyjutTj KOL ypafjifxr] Z*], e7r€t rj ypaixfxr] eiTiTTebov^ ahvvarov ra dprjfji^va elvaL^ f For if the points form a series without
lines from a difference in the mode of contact of their points. And so the theory that Hnes consist of points in contact breaks down : for it cannot account for the difference between straight and curved.'
In ^25 one may suspect some corruption in the text. The MSS. read ovK earai 8r] ypafxfxq €k rqs <Tvv(i\lAea>s. The sense required is given in Rota's translation — ' non fiet ex punctorum contactu linea circularis et recta.'
' ^^26-31. The writer has shown that the points, of which the line is supposed to consist, cannot be regarded as united (a) by a-uvexeiu, {b) by avv6eaiSf nor (c) by d(f)r}. He now argues against {d) the view that they constitute * a series ', that they are united by t6 ic^e^ris. (Cf. above, note on 971"^ 16.) He urges here that, whatever may be the case with some ' series \ the series 0/ points must be a series whose members are contiguous, since otherwise they would not form a co7itiniium — i. e. they would not form a line. It appears from Phys. 227* 17-23 that all continua must have their parts ' in contact ' : and all things ' in contact ' must be ecfx^rjs. But there may be to e(p€^r]s without * contact' (e.g. the numerical series), and there may be 'contact' without the contiguous plurality constituting a contitiuuin.
In 11. 29-31 I read as follows : — ro hk (tw^x^^ ovhlv aWo \4yofxev ^ to <| toi; €<tt\v aiTToiiivodv' S}(TT€ Kill ovT(os avdyKT] Tus aTiyfias aiTTeadaij j] [MSS. fj] eivai ypafifxf}v crvvexr].
The clause ro hi (rui'e;^es . . . dnro^ivoiv is direct, and does not depend on ft in 1. 28. The 6e is resumptive. Koi ovtcos, viz. even supposing that the points are f ^c^ijy.
fj iivai ypapuqv avvex^j viz. rj avdyKT] i(TT\v divai kt\.
The meaning concealed in the corrupt ro e^ toy iarh cnrTOfxevcov is rightly given by Rota, ' quod est ex se tangentibus compositum.*
^ ^1-3. The text is here hopelessly corrupt. Apelt conjectures ert, d (iTOTTOv aTiyfxr)v eTTi aTlyixrjs elvcu ^ ypnfxjxrjv koL cttI ariyixrjf, eVt de ypay^iqs eirmebov kt\. : and iy.prolegg.^ p. xxii) interprets ' si fieri nequit ut puncto iuxta positum punctum adiungatur, quatenus ne Hnea quidem puncto iuxta posita adiungi potest neque planum lineae . . .' But I do not see how he could defend this translation of his Greek : nor do I see how 972* 3-6 connect with this opening sentence. In his German translation
AR DLI D
972^ DE LINEIS INSECABILIBUS
contact, the line will be divided not at either of the points, but between them : whilst if they are in contact, a line will be the place of the single point. And this is impossible.^ c (j) Further, all things would be divided, i.e. be dissolved, into points ; and the point would be a part of a solid, since the solid — on the theory — consists of planes, the plane of lines, and the lines of points. And since those constituents, of which (as their primary immanent factors) the various groups of things are composed, are 'elements', points would be
* elements ' of bodies. Hence ' elements ' would be identical in nature as well as in name, and not even specifically different.'^
i2 § 5. It is clear, then, from the above arguments that a line does not consist of points.^
(a) But neither is it possible to subtract a point from a line. For, if a point can be subtracted, it can also be added. But if anything is added, that to which it was added will be bigger than it was at first, if that which is added be such as to coalesce and form one whole with it.* Hence a line will be bigger than another line by a point.^ And this is impossible.
But though it is not possible to subtract a point as such from a line, one may subtract it incidentally^ viz. in so far as a point
he proposes to read av i] ypafififj koI fVi o-Tiyfirj, which he translates * wenn auch eine Linie auf einem Punkte sein kann': but one may envy, with- out wishing to imitate, this free-and-easy attitude to Greek Grammar. It seemed best to own myself defeated, and simply to print the original Greek.
^ *3-6. For the argument, cf. above, 971* 28 ff. But what bearing has this dilemma (etre yap . . . ei^' anrovTui) on the preceding lines ?
2 »6- 1 1. In 1. II I read with Apelt, after the MS. \V% ovd" erepa^ for Bekker's ov^erepa.
The common name ' o-toix^Iov ' would indicate a genuine identity of nature in the different things called * elements': indeed, complete identity of nature, and not merely generic identity with specific differences.
In 1. 10 eKaara means, of course, not each thing, but each group or kind of thing.
^ The writer has shown that a line is not in any sense a S2wi of points.
He now shows that you cannot speak of sicbtracti7ig a point from a line :
and from this proceeds to criticize other erroneous statements about
* points *.
* In *I5 the MSS. read to irpoaredev (to npoaTcBrjTO) L) n^l^ov earai tov
Apelt conjectures to a> Trpoa-eredr] jjiflCov /ctX., and this seems undoubtedly right. The corruption may have arisen from the mistaken assumption that TOV f^ apxrjs means * than the original quantum '.
° In *I7 I read with Hayduck eartt {(tpa) ypafifirj kt\.
SECTION V 97a'
is contained in the line which one is subtracting from another line. For since, if the whole be subtracted, its beginning and 20 its end are subtracted too ; and since the beginning and the end of a line are points : then, if it be possible to subtract a line from a line, it will be possible also thereby to subtract a point. But such a subtraction of a point is incidental or per accidens} 24
(b) But if the limit touches that of which it is the limit (touches either it or some one of its parts), and if the point, qnd limit of the line, touches the line, then the line will be greater than another line by a point, and the point will consist of points. For there is nothing between two things in contact.-
The same argument applies in the case of division, since the 28
* division ' is a point and, qua dividing-point, is in contact with something. It applies also in the case of a solid and a plane. And the solid must consist of planes, the plane of lines, just as (on the theory) the line consists of points.^
' »20-24. I follow Hayduck and Apelt in reading « (ya/j) rov oXou a.<}>aipoviJL€Uov Koi r) apx>) K.ni to nepas dcfiaipeiTaL, ypa/JLfiijs 5' ^v r) ap)(r] icai ro irepas aTiyfir], kol d ypnpfirjs (^ypapp,r]v) iyxo^p^ d(f)Mp€lv, koI (TTiyixrjv {^avy fvbixoiTo.
^ ^24-27. The writer shows that it is wrong to conceive the limit as ' in contact ' with that which it limits, and the point as * in contact * with the line or any part of it.
In 1. 24 I read (with Apelt) ov to nepas for the MSS. ouVf nepas.
In 1. 25 I punctuate . . . €K€ivov tlvos, rj Se o-Tiyfirj, 7] nepas, ypafxprjs «7rT€Tai, and in 1. 26 I adopt Apelt's conjecture r) peu ovv {ypapprj) ypafx/xTJi i'lTTai (TTiyfxf] pd^av for the MSS. 17 p^v ovv ypapprjs earat ariyprj p€i((op [N 17 p(V ovv ypap/jLi) eaTai (TTiyp^s /leifcoi^].
If the point C becomes the limit of the line AB^ and is therefore *in contact ' with AB, then (i) BA + C is > BA .
by the point C, and (ii) the terminal point ^ I f^ g
C of the line CAB is the co77iposite point
C-\-A : for C and A are in contact whole-with-whole, and there is nothing between them.
^ ^28-30. This passage is obscure owing to its brevity. In 1. 28 I read (with NW^) 6 (5') avTo? Xoyos- . . ., but perhaps we ought to retain the asyndeton, in spite of its harshness. The writer's style, especially at the end of the treatise, is abrupt and compressed in the extreme. In 1. 28 I read <l r) Topx] a-Tiypq [so Z^ : the other MSS. read oTiyprji] koI, 17 [MSS. ^] roprj, anTeTai tivos, and in 1. 30 I accept Apelt's conjecture koi (to eViTreSoj/) €k ypappoov.
If a line consists of points in contact, division of a line— the actual
* cut ' — is itself a point, and {^ud dividing-point) is in contact with the adjacent points, or halves of a point, which it separates. But if so, we shall be led to the same absurdities as before (cf. 972"' 24-27). Hence
D 2
972^ DE LINEIS INSECABILIBUS
30 (c) Neither ^ is it true to say of a point that it is ' the smallest constituent of a line '.
(i) For if it be called ' the smallest of the things contained in the line ', what is ' smallest ' is also smaller than those things 972^ of which it is the smallest. But in the line there is contained nothing but points and lines : and the line is not bigger than the point, for neither is the plane bigger than the line.^ Hence the point will not be the smallest of the constituents in the line.2 4 (ii) And if the point is comparable in magnitude with the line, yet, since ' the smallest ' involves three degrees of com- parison,* the point will not be the smallest of the constituents of the line : or ^ there will be other things in the length besides
we must not regard division as ' dividing a point ', or as itself a ' point of dividing'. But if not, how can a line— which ex hypothesi is nothing but * points in contact ' — be ' divided ' ?
The writer then briefly reminds us that, if a line consists of points in contact, on the same principle a plane is a sum of lines, a solid a sum of planes, in contact with one another : and if we thus conceive solids and planes, 'the same argument' will apply to them. One plane, e.g., will be greater than another by a line, one solid greater than another by a plane, if we are able to ' subtract ' a line from a plane, and a plane from a solid ; and we shall get into difficulties with ' division '.
1 a^o ff. We have seen that we must not predicate ' contact ', ' addition ', ' subtraction ', or ' division ' of the points in a line. In the following arguments the writer shows that we must not say of a point that it is ' the smallest constituent of a line '. No doubt he is attacking a current definition.
2 a^o-l^^. The MSS. read ovk dXrjdes Be Kara crTiyixqv (Indv, old' on r<> eXciXK^TOV [eXaxto'TT] L, Ka\ eAa;^iOT»; P, K«i eXaxiarTov W^J tmv (k ypayip,r]s els TO eXaxia-Tov [to om. L] Ta)V evvTTopxoiTcov e'lprjrai. to 8e eXdxiCTTOv ktX.
The reading, which I have translated, is based on suggestions of Hayduck and Apelt : but I have altered Apelt's punctuation, and sub- stituted y' for de in 1. 33. I read the whole passage thus :—ovk dXrjdei di Kara aTiyfjLrjs etVetr, old' oti to eXdxiCTTov tmv iv ypnfxpLj]. el yap to eXdxtCTTOv Tcov evvTrapxdvTcov e'iprjTai, to y eXux^O'TOU, hu ((TtIv eXdxi-(TT0V, Kai eXciTTuv ecTTiv. ev be tJ} ypapfifi ktX.
* ^2-4. The writer assumes that ///e other constituents of the line, i.e. those presupposed in calling the point 'the smallest' constituent, are infinitesimal (' indivisible ') lines : and the point is not smaller than these. The words in 1. 3, olhe yap av TO emnebov tt^s ypapfxrjs, are obscure. Pre- sumably we are to suppose that (according to the theoiy) just as the line consists of infinitesimal lines = points, so the plane consists ofptanes- o/-z9i^m'teszma/-^readt/t = \\nes.
* ^5. ev Tpial TTpoa-MTTois. The word does not appear to be used in this sense elsewhere in Aristotle.
° ^'6. I read . . . tmv ev tjj ypnp.nrj eXdxKTToi; {^) Koi nXX' r'trra euvTrdp^ei [so Hayduck for the MSS.ei'virdpxei] Trnpa ktX. The insertion of rj seems
SECTION V 972
the points and lines, so that it will not consist of points.^ But, since that which is in place is either a point or a length or a plane or a solid, or some compound of these : and since the constituents of a line are in place (for the line is in place) : and since neither a solid nor a plane, nor anything compounded of 10 these, is contained in the line : — there can be absolutely nothing in the length except points and lines.^
(iii) Further, since that which is called ' greater ' than that which is in place is a length or a surface or a solid: then, since the point is in place, and since that which is contained in the length besides points and lines is none of the aforementioned : — the point cannot be the smallest of the constituents of a length.^
(iv) Further, since ^ the smallest of the things contained in ^7 a house ' is so called, without in the least comparing the house wath it, and so in all other cases : — neither will the smallest of the constituents in the line be determined by comparison with
to be required by the logic of the passage. The writer propounds a dilemma :—
(1) If there are only two kinds of constituent in the line, one of those kinds (viz. the point) cannot be the ' smallest ' ;
(2) If, on the other hand, there are more than two kinds of con- stituent in the line, there must be something other than points and lines contained in it. This he shows to be impossible in the following argument.
^ ^S. I read ovk ap ck o-Tiyfxwv. If the MSS. reading (ov yuf) . . .) be retained, we must translate * For, on this supposition, it will no longer consist of points '.
In ^7 tS ixTjKCL is substituted for rfjypannf.. ypcipurj is determinate /zr/Koy, €7n(f)dveia 'determinate ttXcitos, and aiofxa determinate (Bddos, according to Arist. Met. 1020^ 13.
'^ ^8-13. In 11. 8, 9 I read with Hayduck el de t6 iv totto) ov rj
(TTiyfjif} ^ prjKos [MSS. T) (TTiypi] /x^kos] t} i-nliv^bov f) o-repeov (17) €K tovtwv Ti . . .
^ ^13-17. In ^^14 I read with Hayduck ^ ini.^dvua r) o-Tepeov for the MSS. T) €ni({)dv€ia arfpeov.
The argument is : — The point is ' in place ', i. e. a spatial thing. What is greater than the point, therefore, must be either a line or a plane or a solid. Now, in a length there can be contained neither plane nor solid. Hence there can at most be contained in a length one order of spatial thing (viz. line) which is greater than the point. Hence we are at most entitled to apply the comparative (' smaller'), and not the superlative (' smallest '), to the point in relation to the other constituents of the line.
It is possible, I think, that we should excise et in ^13, and read en xov
fV TOTTCO KtX.
972^ DE LINEIS INSECABILIBUS
the line. Hence the term ' smallest ' applied to the point will not be suitable.^
21 (v) Further, that which is not in the house is not the smallest of the constituents of the house, and so in all other cases. Hence, since the point can exist per se, it will not be true to say of it that it is 'the smallest thing in the line'.^
25 (d) Lastly, the point is not an ' indivisible joint *.^
For (i) the joint is always a limit of two things, but the point is a limit of 07ie line as well as of two. Moreover (ii) the point is an end, but the joint is more of the nature of a division. Again (iii) the lin6 and the plane will be 'joints' (too) : for they are analogous to the point. Again (iv) the joint is in a sense on account of movement (which explains the verse of Empedo- cles ^) : but the point is found also in the immovable things.^
(v) Again, nobody has an infinity of joints in his body or his hand, but he has an infinity of points.^ (vi) Moreover,
31 there is no joint of a stone, nor has it any : but it has points.
^ ^17-21. In ^18 I read /z^ tl rrjS olKias av/i/SaXXo/xcJ/f;? Trpos avro Xeyeriu.
The MSS. give /x/yXf rrjs ktX. Hayduck proposed fifj Trjs, and Apelt
conjectured fxrjre (^npos ti]v olKiav (TVixfidWeTai ftf/re) r^ff oIkius ....
In ^21 I follow Apelt in reading eXdxKTTov. en el for the MSS. iXaxitrrov, enei [eVt P]. . . .
The writer seems to be meeting a possible objection. For it might be said : * It is mere pedantry to object to the superlative. All we meant was that the point is smaller than the infinitesimal lines, or at any rate than the whole line.'
2 ^21-24. 1 read this passage as follows : — en ei t6 nt) ov iv rfj oIkIo. jxt] fCTTi TQ)v iv rfj oIkIo. iXdxKTTov, ofxoicds de koi inl tmv aXXcov, ivbex^rai di [so NW^Z*: the other MSS. read yap] o-Tiyfxrjv avrqv KaO' avrrjv elvai, ovk ecTTai Kara ravTTjs oXTj^e? elTre'iv on to iv ypapmfj iXax^crrov. 'in 6' ovk ktX. [So Hayduck and Apelt : the MSS. read iXaxia-rov, on de ovk, or on ovk.]
The writer criticizes the definition on the ground that it assumes that the point is esse7itially a constituent of a line, i. e. has no being except in a line.
' ^25. We must not describe the point as * an indivisible joint '. We do not know who thus described it, but no doubt the writer is attacking a current description.
■* ^27-31. I read dvoKoyov yap ^xovaiv. en [so Apelt, following W* ; the other MSS. have ixovaiv, 6n] t6 apdpov 8id (f)opdv [so Apelt, for the MSS. dia(})opd or bidcjiopov] nws iariv ....
What the verse of Empedocles was, is unknown : the MSS. give * 5*6 del 6p6a>s% for which Diels {Vorso^^^atiker, 2nd ed., vol. I, p. 184) brilliantly conjectures Sua) ficfi lipOpov, ' the joint binds two '.
'' ^30. The MSS. have rj de anypr] Ka\ to iv toIs aKiv^Tois. The to is unintelligible, and Hayduck is no doubt right in excising it.
" ^31. The MS. L exhibits (TT6p.an for o-w/xan in its margin. But this looks like a correction. The argument is a fortiori. * In one's body— nay, even in one's hand— there are an infinity of points. . . .'
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